Maybe the best way to approach this question is to study $F^*(G)$, the generalized Fitting subgroup of $G$. This splits into two parts:

• $F(G)$ -- the Fitting subgroup. This is a direct product of $p$-groups, and your condition requires that they all have exponent $p$. The theory of such $p$-groups is extensive and there are plenty of examples: elementary abelian groups, extraspecial groups, etc.

• $E(G)$ -- the Layer. This is a direct product of quasisimples. As I said in the comment above, your condition implies that the Sylow $2$-subgroups are elementary abelian so each of these quasisimples $T$ will have $T/Z(T)$ isomorphic to one of the groups listed in Walter's theorem. These are $PSL_2(q)$ with $q\equiv 3,5\pmod 8$, or else with $q$ even; $J_1$ (one of Janko's sporadic groups); ${^2G_2(3^{2a+1})}$ (one of the families of Ree groups). Your condition is stronger than Walter's, so some of these can be excluded (for instance, in the $PSL(2,q)$ case, your condition requires that $q(q^2-1)$ is not divisible by an odd square). But, anyway, you could check these.

So there are plenty of possibilities for $F^*(G)$ and these should be classifiable (at least up to the classification of $p$-groups of exponent $p$). Now Bender's famous theorem about $F^*(G)$ -- that $C_G(F^*(G)) = Z(F^*(G))$ -- tells you the possibilities for the group $G$ itself (at least in theory).

To obtain a complete answer using Bender's theorem, you would need to know the structure of the automorphism group of all parts of $F^*(G)$ -- this is fine for the quasisimples involved in the layer, but more tricky for the $p$-groups part. You'd also need to consider the original question for subgroups of the symmetric group (to deal with the situation where the layer contained multiple copies of the same quasisimple).

Maybe the best way to approach this question is to study $F^*(G)$, the generalized Fitting subgroup of $G$. This splits into two parts:

• $F(G)$ -- the Fitting subgroup. This is a direct product of $p$-groups, and your condition requires that they all have exponent $p$. The theory of such $p$-groups is extensive and there are plenty of examples: elementary abelian groups, extraspecial groups, etc.

• $E(G)$ -- the Layer. This is a direct product of quasisimples. As I said in the comment above, your condition implies that the Sylow $2$-subgroups are elementary abelian so each of these quasisimples $T$ will have $T/Z(T)$ isomorphic to one of the groups listed in Walter's theorem. These are $PSL_2(q)$ with $q\equiv 3,5\pmod 8$, or else with $q$ even; $J_1$ (one of Janko's sporadic groups); ${^2G_2(3^{2a+1})}$ (one of the families of Ree groups). Your condition is stronger than Walter's, so some of these can be excluded. (For instance the Ree groups have elements of order $9$ and so can be excluded; similarly, in the $PSL(2,q)$ case, your condition requires that $q(q^2-1)$ is not divisible by an odd square.) But, anyway, you could check these.

So there are plenty of possibilities for $F^*(G)$ and these should be classifiable (at least up to the classification of $p$-groups of exponent $p$). Now Bender's famous theorem about $F^*(G)$ -- that $C_G(F^*(G)) = Z(F^*(G))$ -- tells you the possibilities for the group $G$ itself (at least in theory).

To obtain a complete answer using Bender's theorem, you would need to know the structure of the automorphism group of all parts of $F^*(G)$ -- this is fine for the quasisimples involved in the layer, but more tricky for the $p$-groups part. You'd also need to consider the original question for subgroups of the symmetric group (to deal with the situation where the layer contained multiple copies of the same quasisimple).