Let A be a finite dimensional symmetric k-algebra over some field k. The set of units of A is denoted by U(A). Suppose G is a cyclic group of prime order which acts via inner algebra automorphism on A, say, there is a homomorphism $\phi : G \rightarrow U(A)$ such that $a\cdot g:= a ^{\phi(g)}$ for all $a\in A, g\in G$ and this action preserves scalar multiplication by k. We donote the fixed subring under the action of G by $A^G$. My question is the following:

Is $A^G$ always a symmetric k-algebra in this condition?