Semidirect products and invariant connections

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 $N$.