  We introduce some special presentations on finite groups, that we call
"diagonal double Kodaira structures" and whose existence is equivalent to the
existence of some special Kodaira fibred surfaces, that we call "diagonal
double Kodaira fibrations". This allows us to rephrase in purely algebraic
terms some results about finite Heisenberg groups, previously obtained in the
recent work of the author with A. Causin, and makes possible to extend them to
the case of arbitrary extra-special $p$-groups.
