Let $V$ be a representation of $G$ and $W$ a subrepresentation. Let $\pi:V\to W$ be an arbitrary projection, and let $$ \pi'(v)=\frac 1{|G|}\sum_{g\in G} g^{-1}\pi (gv) $$ This map is obviously $G$ -equivariant, and is the identity on $W$ , and its image is contained in $W$ , since $W$ is invariant under $G$ . Thus it is an equivariant projection to $W$ , and its kernel is a complement to $W$ .