Let $\mathfrak{g}$ and $\mathfrak{h}$ be finite dimensional complex Lie algebras. Two Lie algebra homomorphisms $\rho$ and $\phi$ of $\mathfrak{g}$ into $\mathfrak{h}$ are said to be equivalent if there exists a Lie algebra automorphism $\psi$ of $\mathfrak{h}$ such that $$\rho = \psi \circ \phi.$$
Let us denote the set of equivalence class of the above equivalence relation on the set of all Lie algebra homomorphisms of $\mathfrak{g}$ into $\mathfrak{h}$ by $\operatorname{hom}(\mathfrak{g},\mathfrak{h})/\operatorname{Aut}\mathfrak{h}$.
If $\mathfrak{h_m}$ denotes the complex Heisenberg Lie algebra of dimension $2m + 1$ and $\operatorname{Der}(\mathfrak{h}_m)$ is the Lie algebra of all the derivations of $\mathfrak{h}_m$, I would like to know what is generally known about
$$\operatorname{hom}\big(\mathfrak{sl}(2,\mathbb{C}),\operatorname{Der}(\mathfrak{h}_m)\big)/\operatorname{Aut}\big(\operatorname{Der}(\mathfrak{h}_m)\big).$$
Other question: Is there a classification of the Lie algebras $\mathfrak{sl}(2,\mathbb{C}) \ltimes_{\rho} \mathfrak{h}_m$ (up to isomorphism of Lie algebras)?
Thanks in advance.
[Note: Your “other question” is essentially a restatement: indeed, it consists of also modding out by precomposition by the automorphism group of $\mathfrak{sl}_2$, but since the latter has only inner automorphisms this changes nothing.]
In general, if $\mathfrak{s}$ is semisimple and $\mathfrak{h}=\mathfrak{l}\ltimes\mathfrak{r}$ ($\mathfrak{r}$ solvable radical, $\mathfrak{l}$ a Levi factor), the canonical map $\operatorname{Hom}(\mathfrak{s},\mathfrak{l})\to \operatorname{Hom}(\mathfrak{s},\mathfrak{h})$ (given by composition with the canonical projection) induces a bijection $\operatorname{Hom}(\mathfrak{s},\mathfrak{l})/\operatorname{Aut}(\mathfrak{l})\to \operatorname{Hom}(\mathfrak{s},\mathfrak{h})/\operatorname{Aut}(\mathfrak{h})$.
In your case, the Levi factor in the Lie algebra of derivations of $\mathfrak{h}_m$ is the symplectic Lie algebra $\mathfrak{sp}_{2m}$. If I’m correct, $\operatorname{Hom}(\mathfrak{sl}_2,\mathfrak{sp}_{2m})/\operatorname{Aut}(\mathfrak{sp}_{2m})$ is the same as the set of $2m$-dimensional representations preserving a symplectic form (there’s a little verification to check that 2 such representations are linearly equivalent iff they are equivalent by an isomorphism of symplectic spaces). Again, if I’m correct, a representation of $\mathfrak{sl}_2$ preserves a symplectic form iff the multiplicity of every odd-dimensional irreducible is even.
Hence, the number of equivalences classes of homomorphisms $\mathfrak{sl}_2\to\operatorname{Der}(\mathfrak{h}_m)$, or also of semidirect products $\mathfrak{sl}_2\ltimes\mathfrak{h}_m$ is the number of partitions of $2m$ in which each odd number occurs an even number of times (Sloane A015128): for $m=1,2,\dots,10$ this is $1, 2, 4, 8, 14, 24, 40, 64, 100, 154$. Among these, there is the 1-component partition corresponding to the irreducible $2m$-dimensional representation, and the discrete partition, which corresponds to the trivial representation (corresponding to the direct product $\mathfrak{sl}_2\times\mathfrak{h}_m$).