Let $F$ be a Siegel cusp form of weight $k\geq2$ (integer), degree two for the group $\text{Sp}_4(\mathbb{Z})$. Suppose it is an eigenform for all the Hecke operators $T(n)$. Let $\sigma$ be an automorphism of $\mathbb{C}$. Let $F^{\sigma}$ denote the formal series obtained by letting $\sigma$ act on the coefficients of $F$: i.e., if $$ F(Z) = \sum_{T}A_F(T)e^{2\pi i Tr(TZ)} $$ where $T$ runs over $2\times2$ symmetric, positive definite half-integral matrices, then $$ F^{\sigma}(Z) : = \sum_{T}\sigma(A_F(T))e^{2\pi i Tr(TZ)}. $$
Question(s): Is $F^{\sigma}$ again an eigenform of degree two for the group $\text{Sp}_4(\mathbb{Z})$? If so, is there an easy proof?
My question is motivated from this question regarding modular forms (of degree one), which asks if $f$ is an eigenform for $\text{SL}_2(\mathbb{Z})$, then is $f^{\sigma}$ also an eigenform (which is true in this case).
Edit: Instead of letting $\sigma$ act on the coefficients $A_F(T)$, what will happen if we consider the new 'formal' series obtained by letting $\sigma$ act on the eigenvalues? I am not sure if this makes sense or not, will this be a Siegel form at least?
