Don´t you like my answer? Maybe you need an explanation?

Are there some groups which action is always conjugate to a linear action in $\mathbb R^4$? Which ones?

This is a really genuine question, with the hope someone knows some application, at least in theoretical physics.

This is just the distributive law common!!!

I meant just a morphism from $G$ to $G$ not an automorphism, which is what he is asking for. Sorry, this are just words, my idea is correct is a shame you couldn´t read this from what I wrote.

msp.org/pjm/1978/76-1/pjm-v76-n1-p15-s.pdf this may help.

I guess what you are looking is a continuous family of group automorphisms. Of course this depend on the group and the topology in the set of the automorphisms. If the group is finite then obviusly the set of automorphisms is finite.

Shouldn´t it be accepted?