In this paper, I will start with posing three fundamental and old questions on (elation) generalized
quadrangles, and survey tersely answers on these questions coming from recent work of S. E. Payne and the author of
this paper. I will then introduce a fourth question posed recently by S. E. Payne, and will provide a general answer to
this question, a result independently obtained by R. Rostermundt for the Hermitian quadrangles $H(3,q^2)$, $q$ even, in
an entirely different fashion. Finally, I will show that this answer yields examples of elation generalized quadrangles
for which the automorphism group fixing the elation point is not induced by the automorphisms of the elation group
fixing the associated $4$-gonal family.