Extensions of groups by weighted Steiner loops

Solving functional equations given in \cite{nagy} for extensions of a group $A$ by a weighted Steiner loop $S$ we obtain concrete description for all loops with interesting weak associativity properties if the Steiner loop $S$ induces only the trivial automorphism on $A$. We show that the restricted Fischer groups and their geometry play an important role for loop extension with right alternative property. Also the automorphism groups of these extensions as well as the conditions for isomorphisms between two extensions are studied.