On blocks of an action.

As we know a finite group G acts transitively on each of it's conjugacy classes. I was trying to come up with an example of a nontrivial block in this action, or in other words, an example where the action of G on a conjugacy class is not primitive.

Your inputs would be greatly appreciated!

One way to think of an example is as follows: Consider a group $G$ with a normal subgroup $H$, and let $\langle h \rangle$ be a conjugacy class in $G$ for $h \in H$. Then (sometimes) this conjugacy class breaks up into more than one conjugacy class in $H$. The action of $G$ by conjugation on $\langle h \rangle$ will then be imprimitive with blocks corresponding to the conjugacy classes in $H$ (providing that they have order bigger than one).
An explicit example: suppose that $G = S_p$ and $H = A_p$ where $p > 3$ is prime, and $h = (1,2,3,\ldots,p) \in A_p$. Then the conjugacy class of $h$ in $G$ has size $(p-1)!$, but this breaks up into two conjugacy classes of size $(p-1)!/2$ in $A_p$. This is bigger than one if $p > 3$. Hence the action of $S_p$ has two "blocks" corresponding to the two conjugacy classes under $A_p$.
Another general example. Let $\Gamma$ be any non-abelian group. Let $\gamma \in \Gamma$ be a non-central element. Then let
$$G = \Gamma \wr S_n = \Gamma^n \rtimes S_n,$$
and let $H = \Gamma^n$ with $h = (\gamma,e,e,\ldots,e)$. Then the conjugacy class of $h$ inside $H$ lives inside the first copy of $\Gamma$, but the conjugacy class of $h$ inside $G$ is $n$-times larger