Let $G$ be a complex simple Lie group with Lie algebra $\mathfrak g$ and $G_1$ be a complex simple subgroup of $G$ with Lie algebra $\mathfrak g_1$. We may assume $G$ and $G_1$ to be of adjoint type. Suppose $H_1,H_2 \in \mathfrak g_1$ are two semisimple elements such that $ \ \exists \ \alpha \in G $ such that $\ \alpha H_1 \alpha^{-1}=H_2$. Does there exist $ \ \ \beta \in G_1 $ such that $\ \beta H_1 \beta^{-1}=H_2$?
This is not true if we assume $G_1$ to be a real subgroup.
In the direction of a positive answer to your question, there are some rather restrictive hypotheses which would suffice, though I'm still not optimistic about getting a positive answer in your very general situation. Note that $G = \mathrm{GL}(V)$ might as well be a general linear group (or other reductive group), since the center doesn't affect conjugation or the adjoint action on the Lie algebra.
If $G_1$ is embedded in a general linear group $G$ by an irreducible representation (say of algebraic groups), then you can appeal to known results to conclude that conjugacy (= similarity) of two semisimple elements of the Lie algebra $\mathfrak{g}_1$ under $G$ for all such representations implies conjugacy under the adjoint action of $G_1$ on $\mathfrak{g}_1$. But having to consider all irreducible representations of $G_1$ gets outside your framework even if you start with a nice embedding.
For Lie algebras this is a characteristic 0 theorem, proved by M. Gauger here. Soon afterward, Steinberg streamlined the argument and also obtained a close analogue for the groups with less restriction on the characteristic here.
In any case, the algebraic group setting of these papers is the most natural one for dealing with semisimple elements of the groups or Lie algebras.
