Characterizing pyramidal Hadamard designs with the largest number of fixed points

A symmetric $(v,k,λ)$-design is said to be $f$-pyramidal, with $f<v-1$, under the action of a group $G$ if $G$ acts as an automorphism group fixing $f$ points and acting sharply transitively on the remaining ones. We show that, necessarily, $f\leq v -2(k-λ)$. In particular, for a Hadamard design wit