The endomorphism rings of permutation modules of $\frac{3}{2}$-transitive permutation groups

Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with six families of groups which are $2$-transitive, or Frobenius, or one-dimensional affine, or the affine solvable subgroups of $\mathrm{A}\mathrm{G}\mathrm{L}(2,q)$, or special projective linear group $\mathrm{P}\mathrm{S}\mathrm{L}(2,q)$, or $\mathrm{P}\mathrm{\Gamma }\mathrm{L}(2,q)$, where $q=2^p$ with $p$ prime. According to a case by case analysis, we prove that the endomorphism ring of the natural permutation module for a $\frac{3}{2}$-transitive permutation group is a symmetric algebra.