Let $P$ be a group with normal subgroups $G$ and $H$, with $G \not \subset H$, $H \not \subset G$ and $G \cap H \neq 1$. Consider group actions $\theta : G \to Aut(H)$ and $\xi: H \to Aut(G)$ such that the restrictions $\theta (g) |_{G \cap H}$ and $\xi (h) |_{G \cap H}$ acts like conjugations, for all $g \in G$ and all $h \in H$, where $Aut( \cdot )$ is the group of automorphisms of the argument. Suppose that both $\theta$ and $\xi$ do not act like conjugations outside $G \cap H$.
Someone could show some example of groups and actions like those?
Thanks in advance
Take $P=\mathbb{Z}/12\mathbb{Z}$, $G=\langle \overline{2} \rangle$ and $H=\langle \overline{3} \rangle$. Then $G\cap H = \langle \overline{6} \rangle$ is non trivial, $G$ and $H$ are normal in $P$, and none of $G$ or $H$ is contained in the other.
Now, define an action $\theta$ of $G$ on $H$ as follows: $\theta(\overline{2})$ is the multiplication by $\overline{3}$ in $H$. This automorphism is of order $2$, so it extends to a full action of $G$ on $H$ by $\theta(\overline{2k}) = \theta(\overline{2})^k$. Moreover, this action fixes $\overline{6}$.
Similarly, define an action $\xi$ of $H$ on $G$ as follows: $\xi(\overline{3})$ is multiplication by $\overline{5}$ in $G$. Again, this extends to a full action of $H$ on $G$, and $\overline{6}$ is fixed by this action.
Thus $\theta$ and $\xi$ act trivially on $G\cap H = \langle \overline{6} \rangle$, and this is exactly the action by conjugation, since the group is abelian. But $\theta$ and $\xi$ are non-trivial actions, so they do not act by conjugation, again since the group is abelian.
Therefore all the conditions from the statement are satisfied.