P = PGL(3,4) as a permutation group on 21 points: faithful by construction (60480 distinct permutations).
Q transitive on the 20 other points (P 2-transitive, so Q is maximal): True
phibar(Q) = T (so phi(Q) = Inn(T), T = A_5): True
orders of the normal subgroups of P (own lattice): [1, 20160, 60480] -> orders of the quotients: [60480, 3, 1]
=> no quotient of P has order divisible by 60 and at most 120, so no homomorphism P -> Aut(A_5) = S_5 has an image containing Inn(A_5); phi does not extend to P and Im(phi) = A_5 is not a homomorphic image of P.
subgroups of A_5 generated by two elements (all subgroups of A_5), by order -> centralises some t != 1: {1: [True], 2: [True], 3: [True], 4: [True], 5: [True], 6: [False], 10: [False], 12: [False], 60: [False]} ; number of subgroups: 59
for all 427 pairs (class, orbit) of the 100 classes: good at d <=> induced order <= 5: True
classes of order > 240: 12 ; smallest induced order among their orbits: 6
