Let $G$ and $H$ be two finite groups such that $H$ acts on $G$ through the automorphism $\varphi : H \longrightarrow \text{Aut}(G)$, i.e., $h \cdot g = \varphi_{h}(g) $, where I am writing $\varphi(k) = \varphi_{k}$. Then, one can construct a semi-direct product $G \ltimes H$, which is $\{(g,h):g\in G, h \in H\}$ and the group law is given as: $$ (g_{1},h_{1})\cdot (g_{2}, h_{2}) = ( g_{1}\varphi_{h_{1}}(g_{2}),h_{1}h_{2}).$$ I would like to describe the conjugacy classes of $G \ltimes H$, possibly in terms of those of $G$ and $H$.
In the semi-direct product, the inverse elements are given as $(g,h)^{-1} = (\varphi_{h^{-1}}(g^{-1}), h^{-1})$. Therefore, the conjugacy operation in $G \ltimes H$ is given as $$ (g_{1},h_{1}) \cdot (g_{2},h_{2}) \cdot (g_{1},h_{1})^{-1}) = \big( g_{1} \varphi_{h_{1}}(g_{2}) \varphi_{h_{1}h_{2}h_{1}^{-1}}(g_{1}^{-1}), h_{1}h_{2}h_{1}^{-1}).$$
So, the second coordinate of every conjugacy class $\mathcal{D}$ of $G \ltimes H$ lies in the single conjugacy class $\mathcal{C}$ of $H$. However, it seems entirely possible a conjugacy class $\mathcal{C}$ of $H$ may correspond to two or more conjugacy classes of $G \ltimes H$. I do not know how to bring conjugacy classes of $G$ into the picture, while describing the conjugacy classes of $G \ltimes H$.
Since the semi-direct product is such a well-known construction, I would not be surprised if this has already been studied. I would really appreciate help or if anyone can point me to appropriate references.