Philip Hall's problem on non-Abelian splitters

Philip Hall raised the following question which is stated in The Kourovka Note-book [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 ≠ 1 with 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^ω ≅ G, Hom (L^ω, S_ω) = 0 (S_ω the infinite symmetric group acting on ω) and End (L, L) = Inn L∪ {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.