Multiplicative subgroups are not restricted sumsets

We determine exactly which proper multiplicative subgroups of a prime field can be represented as a restricted sumset of the form $A\mathbin{\widehat{+}} A=\{a+a':a,a'\in A,\ a\ne a'\}$. We prove that a proper multiplicative subgroup $H\le\mathbb F_p^*$ cannot satisfy $H=A\mathbin{\widehat{+}} A$ wh