I am wondering how to construct unitary irreps of the group $$(SO(3) \times SO(3)) \rtimes \mathbb{Z}_2$$ where the $\mathbb{Z}_2$ acts by swapping the two copies. This is the isometry group of $\mathbb{R} P^3 \simeq SO(3)$.
I know that irreps of $SO(3) \times SO(3)$ are given by $V_i \otimes V_j$ where $V_i$ is an irrep of the 1st factor and $V_j$ is an irrep of the 2nd factor. However, I am not sure how the semi-direct product messes this up.
More generally I am interested in unitary irreps of groups like $$ \underbrace{ \left( SO(3) \times \cdots \times SO(3) \right) }_n \rtimes G $$ where $G$ is a subgroup of the symmetric group on $n$ letters. In particular, $G$ acts on the $n$ copies of $SO(3)$ according to its permutation representation.