Let $X$ be a connected (locally Noetherian) scheme and $\phi: Y \rightarrow X$ be a finite etale Galois cover of degree $d$, i.e. $Y$ is connected and there are $d$ $X$-invariant automorphisms of $Y$.
Let us fix geometric basepoints $\overline{y}$ and $\overline{x}$, where $\overline{y}$ lies over $\overline{x}$. We obtain a map $$ \phi_*: \quad \pi_1(Y, \overline{y}) \rightarrow \pi_1(X, \overline{x}) $$ Finally, let us write $F: \text{FEt}/X \rightarrow (\pi_1(X, \overline{x})-\text{Set})$ for the fibre functor.
Question: Is it true that the image of $\phi_*$ consists of exactly those $g \in \pi_1(X, \overline{x})$ that act trivially on $F(Y)$?
More elaborately, let $$ U = \{ g \in \pi_1(X, \overline{x}) \, | \, g \text{ acts trivially on } F(Y) \} $$ Then $U$ is a intersection of the stabilizers of all the points in $F(Y)$, since there are finitely many such points, $U$ is open. Does the following hold? $$ \operatorname{im}(\phi_*) = U $$
Attempt at proof: It is known that given a connected finite etale cover $Z \rightarrow X$, it is trivialized by $Y$ if and only if $\operatorname{im}(\phi_*)$ acts trivially on $F(Z)$. Since $Y$ is trivialized by itself (it is Galois), we see that $\operatorname{im}(\phi_*)$ acts trivially on $F(Y)$, hence $\operatorname{im}(\phi_*) \subseteq U$. For the other inclusion, I have really no idea how to proceed.
Background: In Szamuely's Bourbaki report "Corps de classes des schémas arithmetiques" he states that (using the same notation and writing $G = \operatorname{Aut}_X(Y)$) the quotient of $\pi_1^{\text{ab}}(X)$ by the image of $\pi_1^{\text{ab}}(Y)$ is $G^{\text{ab}}$. Note that here everything is abelianized. This question here is my attempt to prove this claim: if the answer to my question is positive we would know that $G$ is the group opposite to the quotient of $\pi_1(X, \overline{x})$ by the image of $\pi_1(Y, \overline{y})$ and then the result would follow by using functoriality and right exactness of abelianization.