17.95 (2010)

Let $G$ be a permutation group on the finite set $\Omega$. A partition $\rho$ of $\Omega$ is said to be $G$-regular if there exists a subset $S$ of $\Omega$ such that $S^g$ is a transversal of $\rho$ for all $g \in G$. The group $G$ is said to be synchronizing if $|\Omega| > 2$ and there are no non-trivial proper $G$-regular partitions on $\Omega$.
a) Are the following primitive groups of affine type synchronizing:
$2^p.\text{PSL}(2, 2p+1)$ where both $p$ and $2p+1$ are prime, $p \equiv 3 \pmod 4$ and $p > 23$?
$2^{101}.\text{He}$?
b) For which finite simple groups $S$ are the groups $S \times S$ acting on $S$ by $(g, h): x \to g^{-1} x h$ non-synchronizing?