Consider a linear representation of a group $G$ on a ($n$-dimensional) Hilbert space $H$ (complex vector space with sesquilinear inner product $\langle\cdot,\cdot\rangle$ ):
$$ g\in G \mapsto \rho(g)\in B(H),\quad |\psi\rangle\in H \mapsto |\phi\rangle = \rho(g)|\psi\rangle\in H $$
where $B(H)$ is the space of operators/matrices on $H$ and $\rho(g_1g_2) = \rho(g_1)\rho(g_2)$ with $\rho(e) = \mathbb{1}_H$. This representation also induces a natural representation of $G$ acting onto $B(H)$ via conjugation:
$$ g\in G \mapsto \Phi_1(g) \in B(B(H)),\quad O\in B(H) \mapsto O' = \Phi_1(g)O = \rho(g)O\rho(g)^{-1} \in B(H) $$
One can check that this is in fact a linear representation (i.e. $\Phi(g)$ acts as an $n^2\times n^2$ matrix onto the "vectorized" operator $O$), although I don't think that this is in general a faithful representation.
Another natural representation induced on $B(H)$ is obtained via the hermitian conjugate $M\in B(H) \mapsto M^\dagger\in B(H)$ (which may be seen as the complex conjugate of the transpose in a suitable basis):
$$ g\in G \mapsto \Phi_2(g) \in B(B(H)),\quad O\in B(H) \mapsto O' = \Phi_2(g)O = \rho(g)O\rho(g)^\dagger \in B(H) $$
Now this is also a linear representation of $G$ on $B(H)$ considered as a vector space. Is there a name which identifies it? Are there general known results one should be aware of? I'm especially interested in the case where $G$ is a (finite-dimensional, possibly reductive) Lie group.
Is there a relationship between the "conjugation" representation $\rho(g)(\cdot)\rho(g)^{-1}$ and the "hermitian conjugate" representation $\rho(g)(\cdot)\rho(g)^\dagger$? Namely, are they the same representation (up to isomorphism)?