Character of a Hom representation
= Character of a Hom representation
For complex representations $V,W$ of a finite group,
$$
\operatorname{Hom}_{\mathbb C}(V,W)\cong V^*\otimes W.
$$
Under the $G\times G$ action $(g,h)\alpha=\sigma(h)\alpha\rho(g^{-1})$, its character is
$$
\chi_{\operatorname{Hom}(V,W)}(g,h)
=\overline{\chi_V(g)}\,\chi_W(h).
$$