  In this paper, we study geometric properties of the set of group invariant
continuous linear operators between Banach spaces. In particular, we present
group invariant versions of the Hahn-Banach separation theorems and elementary
properties of the invariant operators. This allows us to contextualize our main
applications in the theory of norm-attaining operators; we establish group
invariant versions of the properties $\alpha$ of Schachermayer and $\beta$ of
Lindenstrauss, and present relevant results from this theory in this (much
wider) setting. In particular, we generalize Bourgain's result, which says that
if $X$ has the Radon-Nikod\'ym property, then $X$ has the $G$-Bishop-Phelps
property for $G$-invariant operators whenever $G \subseteq \mathcal{L}(X)$ is a
compact group of isometries on $X$.
