I contemplated this question a bit for smooth projective varieties (there are interesting such examples with conjugates with distinct fundamental groups; in addition to the examples of Serre, see more recent examples of Stover). I don't know much algebraic geometry, so I may be saying some things that are incorrect. The upshot is that this should hold for smooth complete intersections in abelian varieties.

The first reduction is to consider varieties with fundamental group $\mathbb{Z}^{2n}$ by passing to a finite abelian cover. For $n=0$, one has for example projective space $\mathbb{P}^k$, and for $n>0$, one has abelian varieties. These spaces should have Galois conjugates having fundamental group $\mathbb{Z}^{2n}$, e.g. products of elliptic curves.

Next, I think one can reduce to the case of algebraic surfaces. By the Lefschetz hyperplane theorem, the intersection of a smooth projective variety $X^m \subset \mathbb{P}^k$, $m=dim X > 2$ with a generic (transverse) hyperplane $\mathbb{P}^{k-1}\subset \mathbb{P}^k$ will have intersection $X^m \cap \mathbb{P}^{k-1} = X^{m-1}$ a smooth subvariety of one less dimension $m-1$ and with $\pi_1(X^{m-1})\cong \pi_1(X^m)$. Iterating, we see that there is a surface $X^2\subset X^m$ such that $\pi_1(X^2)=\pi_1(X^m)$ such that $X^2=X^m\cap \mathbb{P}^{k-2} \subset \mathbb{P}^k$.

However, there doesn't seem to be much known about surfaces with abelian fundamental group, even simply-connected.
I think one case that follows though is the case of surfaces which are complete intersections of smooth hypersurfaces intersecting transversely (see Mandelbaum). In fact, Bott's argument shows that any transverse holomorphic section of a positive line bundle over a variety (smooth ample divisor) satisfies the Lefschetz theorem, and hence will be simply connected. But the property of being a transverse intersection of smooth ample divisors should be a Galois-invariant property, and hence the Galois conjugates $\sigma X$ should also be simply-connected.

More generally, one gets smooth projective manifolds with abelian fundamental group by taking a abelian variety $A$ and crossing it with projective space $A \times \mathbb{P}^k$. Again, this property should be invariant under Galois conjugation, since abelian varieties admit an algebraic group structure which is Galois-invariant. Now take sequences of smooth ample divisors on such spaces to obtain smooth projective surfaces with fundamental group $\mathbb{Z}^{2n}$ and whose Galois conjugates should also have abelian fundamental group $\mathbb{Z}^{2n}$ by a similar argument.

So the question one can ask: does every smooth projective variety $X^m$ with $\pi_1(X^m)=\mathbb{Z}^{2n}$ admit a smooth subsurface $X^2\subset X^m$ which is a transverse complete intersection of smooth ample divisors in an abelian variety $A \times \mathbb{P}^k$? Or maybe $X$ itself has this property? This appears to be false for the case of simply-connected smooth projective algebraic surfaces. The thesis of Giancarlo Urzúa mentions that for smooth complete intersections, $c_1^2(X)\leq 2 c_2(X)$. However, there are known examples of simply connected smooth projective surfaces with $c_1^2(X)/c_2(X) > 2$, going back to Moishezon-Teicher. One could then investigate whether Galois conjugates of these surfaces are simply-connected, and then take smooth hypersurfaces in these times abelian varieties.

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