In Silverman's Arithmetic of Elliptic Curves, Chapter III, Proposition 4.12, we have the statement that if $E/F$ is an elliptic curve and $\Phi$ is a $\mathrm{Gal}(\bar{F}/F)$-invariant subgroup then the quotient $E/\Phi$ and the corresponding quotient map $\phi: E\to E/\Phi$ are defined over $F$.

Is the converse true? I.e., if $E/\Phi$ is defined over $F$, then is $\Phi$ necessarily $\mathrm{Gal}(\bar{F}/F)$-invariant?