Symmetric $(2^k-1,2^{k-1},2^{k-2})$-designs which are $(2^{k-1}-1)$-pyramidal over abelian groups

A design is called $t$-pyramidal when it has an automorphism group which fixes $t$ points and acts sharply transitively on the remaining points. We determine all symmetric $(2^k-1,2^{k-1},2^{k-2})$-designs which are $(2^{k-1}-1)$-pyramidal over abelian groups.