Edit: I misunderstood the question, but hey- I made a “How to have a Friend” guide by accident! Not bad, me!
Real answer: Introverts will not fake any extroversion when they are at home and comfortable in their skin. They can spend a long time alone, doing something and are not too keen on getting out of the house; even if it is for an obligatory social event or a simple outing.
Step 1: Get your existence acknowledge by the introvert
Unless you are someone who naturally exudes charisma without even trying, the average person (both extrovert and introvert) rarely notices everyone in their vicinity. At the barest minimum, “notice” is basically “there is a human-like figure passing/standing by near me”.
In this state, people rarely make the effort to reach out until they are certain if they want or need to interact with others.
The introvert will dial this up to the next level. At extreme, they just refuse to do anything with anyone due to impairing feeling of social anxiety. But you can try to “make your way” into the view.
Normally, it’s much easier if you both are in a similar situation; e.g. both of you are classmates/co-workers, or you are chilling in a cafe. Depending on the situation, you may open with innocuous question:
Example 1: “Excuse me- sorry, I can’t help but to notice that book you are reading just now. It looks interesting from the cover- what is it about?”
Example 2: “Hey, are you still looking for a group for the class project? I feel like I want to try pairing up with you; is that okay?” (A/N: Lol)
Regardless if you are rejected or not, you have successfully entered the introvert’s view. Some may still remember your face after a long while. Some may actually steal glances at you every now and then; but would not do much because they are unsure on how to proceed.
Step 2: Establish communication line
Start with something small, like saying a simple hello or a nod of acknowledgement every time you meet the introvert. If it can be done daily, then it’s much better.
You may notice that the introvert would slowly respond more with time. Some introverts will take more time than others to open up, so be patient and let the introvert take the time to be comfortable with you.
At this stage, you are going to create a connection by helping the introvert to feel comfortable around you. You are going to notice the many quirks and interests that they display, consciously and subconsciously- and you will need to commit some of them to memory, so you can take actions that remind them that you too acknowledge them as they are.
Step 3: Create a bond
Be involved in their interests. Ask them if they want to join a DnD group, if they like DnD. Tell them you need help in Genshin co-op, if they also play Genshin Impact (and hopefully, the same server as you do)
Be as available as you could handle, in healthy ways of course. Don’t push yourself to do something you hate/dislike because faking it will lead to deep distrust.
Eventually, you bond will get deeper and the introvert will be comfortable enough to speak up their mind for hours end. That’s when you know the introvert very much appreciate you for listening to them.
Step 4: Maintaining the bond
Because plenty of us, both introverts and extroverts, can be horrible at social skills- learn to maintain your connections with the regular “Hello”s and “Hi”s.
Make the effort to wish them on their birthdays and the many relevant holidays. Message them to ask how their day had been going on (at reasonable hours, please). Send them a meme or two, things that remind you of the introvert.
If the introvert is more than comfortable at talking to you, they might pick up the calls you made. It shows that the introvert has made you a priority; maybe not too high a priority, but a priority nonetheless.
Step 5: Re-evaluate
After some time, re-evaluate your friendship with the introvert. Do you enjoy being around with them and look forward to spend time together? Do you dread at the idea of even talking to them?
If it feels like that you put too much effort into the bond, but the introvert doesn’t seem to care; best to step back for a bit. The introvert may be very terrible at socializing, and they haven’t realized it takes two to tango- if they think differently than you do, it’s neither yours or their faults.
People have different understanding of socializing. Manage your expectations to prevent heart breaks (yes, even in platonic friendship you can still experience a heart break). End it respectfully if the bond is becoming toxic (and either both or one is terrible at self-introspection), because you both deserve fulfilling and caring friendships that might not happen between both of you.
If everything goes well, despite having little life bumps here and there; you got yourself a solid bond with an introvert there.
Yeaaa maybe this is a bit too long for a “How to make friends” guide, but this is roughly the basics you can use on anyone. Be sure you know your intentions on why you approach, because unfulfilled expectations always feel horrible to navigate.
Source: The modus operandi my INFJ ex used on me, from friendship to dating to our final break up. As I said, unfulfilled expectation is horrible especially if you feel you put too much effort in it- so use this knowledge well.