Apply rotation to an action

I already feel sorry for asking such a simple question but I feel forced to since all the tutorials are either cryptic to me or do not cover what I’d like to know.

I have a rigged character with some actions. For example, there is the running animation. I duplicated it in the action editor and renamed it. My goal is to rotate the whole armature in this animation, so the actual object is still facing Y+ but the mesh faces and runs to the right, just like in games when you press A or D but still face forward.
I am able to set single keyframes and overwrite the others but that can’t be the way. I also doesn’t work very well.

So is there a way to apply e.g. a rotation of 90 degrees to all the keyframes of an action?

Thanks in advance, have a nice day!

just key the whole armature, as an object.

I could do that but I’d like to alter the animation a bit so the head turns a bit to the right.

Are you trying to turn just the head or the entire character?

A part of the armature. I want to rotate the body and point the head a bit to the left or right. It’s for walking sideways.