Given two $G$-vector bundles $E$ and $F$ over a $G$-space $X$ ($G$ some finite group), I am interested in the vector bundles $E \oplus F$, $E \otimes F$, $\operatorname{Hom}(E,F)$ etc.
I am familiar with the fact that those are again vector bundles - I think a slick way to see this are continuous functors as described eg. by Atiyah - and there are the obvious group actions on those spaces, obtained as generalizations of the representation-theoretic constructions.
However, is there an easy way of seeing that those actions are continuous? I thought that it would be easy given that $G$ is discrete, but it seemed to get messy and I wonder if there's a somewhat uniform approach. (My background on topological groups and group actions is quasi non-existant, unfortunately)