Group Action on Subgroup by Right Regular Representation is not Transitive

Theorem
Let $G$ be a group.

Let $H$ be a proper subgroup of $G$.

Let $*: H \times G \to G$ be the group action defined as:
 * $\forall \left({h, g}\right) \in H \times G: h * g = \rho_{h^{-1}} \left({g}\right)$

where $\rho_{h^{-1}} \left({g}\right)$ is the right regular representation of $g$ by $h^{-1}$.

Then $*$ is not transitive.

Proof
From Group Action on Subgroup by Right Regular Representation it is established that $*$ is a group action.

From Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset:
 * $\forall x \in G: \operatorname{Orb} \left({x}\right) = H x$

where $H x$ is the right coset of $H$ by $x$.

From Coset Spaces form Partition it is apparent that $x H \ne G$ unless $H = G$.

Thus $\operatorname{Orb} \left({x}\right) \ne G$ and the result follows by definition of transitive group action.