Does every layer of an equivariant composite function have to be equivariant?

Question

For example, given a composite function $f(x)=f_1(f_2(f_3(x)))$, if $f(x)$ is equivariant to the group $G$ (e.g. $SO(3)$), then is it necessary for $f_1$,$f_2$,$f_3$ to be equivariant to $G$? Thanks.

Answer 1

If I'm understanding this question correctly, the answer is no. For instance, let $S^1$ act on $\Bbb{R}^2$ by rotation. Define $\phi:\Bbb{R}^2\to \Bbb{R}^2$ by $(x,y)\mapsto (x,-y)$. $\phi\circ \phi=\mathrm{Id}$, which is of course $S^1$-equivariant. However, $\phi$ and the $S^1$ action do not commute. For instance rotate $(0,1)$ by $\frac{\pi}{4}$ radians counterclockwise.