Since the question about
progression to reach the handstand push up
I decided to share some info, from my personal experience (almost 15 years).
First Handstand push-ups requires a minimal hand balancing ability. So before you start training handstand push-ups, you need to be able to do a hand-stand (not walking) for at least 10 seconds. If you cannot do it - I suggest to focus on that first. If you don't know how -> Here is my video tutorial for that: Handstand basic level
Second If you can do a hand-stand, but don't have enough strength to push your self up to hand stand. Then I suggest to start with Negative training which based on eccentric contraction. By negative - meaning is when you start at handstand and slowly go down. At the beginning you can do it with a mattress (in case you fall). Practicing this gradually increasing reps quantity - will build you the needed strength to push up.
Third To be able of doing many push ups in a row, I suggest to train push ups in 4 sets of maximum reps each set. And since most people loose the balance at the highest point of the move (just after they got up). I suggest to start doing those sets next to wall - so if you lose balance you just put one leg to the wall.
Notes: Don't focus on the time it takes you, instead of that just focus on your progression. Try to get every week at least one thing better than previous.
By the way: 2013 I took 3'rd place at international competition for Calisthenics in handstand push ups category and since 2013 until now I'm record holder of my country.
And for a lit inspiration, here is my fun video I made for last new year: HandStand push ups