Which are the finite groups $G$ such that the element orders of $G$ form an arithmetic progression? Several remarks:

$S_3$, $A_4$ and any $p$-group of exponent $p$ satisfy this property.

If $G$ satisfies this property and $p_1<p_2<...<p_k$ are the prime divisors of $n=\mid G\mid$, then $p_2=2p_1-1$, and consequently $(p_1,p_2)\in \{(2,3),(3,5),(7,13),(19,37),...\}$.

If $G$ satisfies this property and there is $a\in G$ such that $o(a)=\exp(G)$ (in particular, if $G$ is nilpotent), then $G$ is a $p$-group of exponent $p$.

My impression is that the non-nilpotent groups whose element orders form a progression are of order $2^\alpha3^\beta$ (and consequently are solvable), but I failed in proving this.