Projecting a sketch on to a cylinder ends up distorted

I’m just learning so there may be a better way to accomplish this. I’ve created a sketch that I want to create a pattern of for the sides a cylinder. To do this I created the sketch, copied it by using rotate around axis, then tried to project onto the cylinder. The sketches that were on the from turned out right, when I rotate the cylinder though, the sketches on the backside are all messed up.

Here’s a short video of what I’m talking about.

Is there a better way to do this? If not, what am I doing wrong causing the distortion on the sketches?

To add, if it changes how I should do this, my end goal isn’t to extrude those sketches, I plan to export this for printing on a 3d printer and the logos will be colored and flush with the surface.