gr.group theory - Is $varphi(n)/n$ the maximal portion of $n$-cycles in a degree $n$ group?

It is true for all primitive groups: The primitive groups of degree n containing an n-cycle were independently classified in

Li, Cai Heng The finite primitive permutation groups containing an abelian regular subgroup.
Proc. London Math. Soc. (3) 87 (2003), no. 3, 725--747. )

and

Jones, Gareth A.
Cyclic regular subgroups of primitive permutation groups.
J. Group Theory 5 (2002), no. 4, 403--407.

They are the groups G such that

-$C_pleqslant Gleqslant AGL(1,p)$ for p a prime

-$A_n$ for n odd, or $S_n$

-$PGL(d,q)leqslant G leqslant PGamma L(d,q)$: here there is a unique class of cyclic subgroups generated by an n-cycle except for $G=PGamma L(2,8)$ in which case there are two.

-$(G,n)=(PSL(2,11),11), (M_{11},11), (M_{23},23)$

All these groups satisfy the bound.

Gordon Royle has pointed out to me that the bound does not hold for elements of order n. The smallest examples which do not meet the bound are of degree 12 and are the groups numbered 263 and 298 in the catalogue of groups of degree 12 in Magma.