Headstand Strength
Try this in Headstand. First enter your normal position and feel where the weight is, how the weight is shared between the head and hands (3 points). Next try to decrease the weight in the head by increasing the weight in the hands, moving the feet and legs towards under-balance will help. Just practicing this simple drill can dramatically improve your control and efficiency. Play with the weight transfer as you bend the knees to meet the elbows and transition into a Crow Pose (Frog Pose or Bakasana).
Increase Core Strength and Body Control For Press Handstand, HSPU and Crow Transitions using the Headstand! Core strength is funny word and brings a lot of confusion to skills like Press Handstand. I definitely don't contract my rectus abdominals when initiating the Press, I think of it more like blowing air into a long balloon that has been folded in half, its the connection between the legs and the torso, the positioning of the hip towards overbalance and the extension of the posterior chain. However you explain it, it's hard to understand when learning. This is the where the Headstand can be beneficial and you can feel the path of the Pike, Tuck or Straddle Press. For the Handstand Push Up the Headstand is the point at which you change direction from the eccentric to concentric (Down then Up). So having strength, control and balance can make the HPSU much more efficient and decrease the chance of falling.
Check out the YouTube video to see the drills in more detail