  Let $S$ be a complex reductive group acting holomorphically on a complex Lie
group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal
compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via
biholomorphisms. We give an explicit description of the isomorphism classes of
$G$-equivariant almost holomorphic hermitian principal bundles on $N$. Under
the assumption that there is a central subgroup $Z= \text{U}(1)$ of $K(S)$ that
acts on $\text{Lie}(N)$ as multiplication through a single nontrivial
character, we give an explicit description of the isomorphism classes of
$G$-equivariant holomorphic hermitian principal bundles on $N$.
