A note on sumsets of subgroups in $\mathbb Z_p^*$

Let $A$ be a multiplicative subgroup of $\mathbb Z_p^*$. Define the $k$-fold sumset of $A$ to be $kA=\{x_1+\dots+x_k:x_i \in A,1\leq i\leq k\}$. We show that $6A\supseteq \mathbb Z_p^*$ for $|A| > p^{\frac {11}{23} +\epsilon}$. In addition, we extend a result of Shkredov to show that $|2A|\gg |A|^{\frac 85-\epsilon}$ for $|A|\ll p^{\frac 59}$.