While building this plane, I set some activation groups as follows:
1 "Turbojets" on
2 "Rocket" off
4 "Gyros" off
With the plane on the runway, I observed the following problems:
A. Activation groups 1 2 & 4 are not named
B. Activation group 1 was initialized in the off state
Point B might need some discussion. I understand that the question of how staging should interact with activation may seem difficult, but it doesn't seem complicated from my perspective. As an experienced SSTO builder (in another game), my opinion is that activation groups should always take priority over staging. If an actvation group controls the activation of jet engines and it's set to 'on' at launch, then those engines should be on at launch.
Activation groups do work as you mention in your last paragraph, the issue here is that it seems like you had your pilot set to primary when you configured the activation groups, but the cockpit when you launched.
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You didn't configure your cockpit activation groups, just the pilot's, so at launch the pilot saw how the cockpit was configured, and copied its config for himself (this copy behavior is something we added because players usually configured just the cockpit and not the pilot, and when trying to go in first person view controls were misconfigured).
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The solution here would be configuring the cockpit in the designer.