Philip Hall's problem on non-Abelian splitters

Philip Hall raised the following question which is stated in The Kourovka Notebook [12, p. 88]: is there a non-trivial group which is isomorphic with every proper extension of itself by itself? We will split the problem into two parts: we want to find non-commutative splitters, that are groups $G\ne 1$ with ${\rm Ext} (G, G) = 1$. The class of splitters fortunately is quite large so that extra properties can be added to $G$. We can consider groups $G$ with the following properties: there is a complete group $L$ with cartesian product $L^w \cong G, {\rm Hom}(L^w, S^w) = 0$($S_w$ the infinite symmetric group acting on $w$) and ${\rm End} (L, L) = {\rm Inn}\, L\cup\{0\}$. We will show that these properties ensure that $G$ is a splitter and hence obviously a Hall group in the above sense. Then we will apply a recent result from our joint paper [9] which also shows that such groups exist; in fact there is a class of Hall groups which is not a set.