Let $X$ be a topological space, $Y\subset X$ a dense $G_\delta$ set, and $A\subset Y$ such that $A$ is comeager in $Y$, i.e. it is comeager in the relative topology. Does it follow that $A$ is comeager in $U$?
In the case that $Y$ is open, this follows from the fact that $A$ is comeager in $Y$ iff $Y\setminus A$ is meager (in the entire space), since we can write $X\setminus A=(X\setminus Y)\cup (Y\setminus A)$, but I'm not sure if this (that is, the first part of the sentence) holds when $Y$ is $G_\delta$.
Does any of the above change when $X$ is assumed to a Baire space (or even Polish)?
Claim. If $N\subseteq Y\subseteq X$ and $N$ is nowhere dense in $Y,$ then $N$ is nowhere dense in $X.$
Proof. Let $U$ be any nonempty open subset of $X;$ we have to find a nonempty open subset of $U$ which is disjoint from $N.$ If $U\cap Y=\emptyset$ then $U\cap N=\emptyset$ and we're done. Suppose $U\cap Y\ne\emptyset.$ Then $U\cap Y$ is a nonempty open subset of $Y.$ Since $N$ is nowhere dense in $Y,$ there is a nonempty set $V\subseteq U\cap Y$ which is open in $Y$ and disjoint from $N.$ Then $V=W\cap Y$ for some set $W$ which is open in $X.$ Finally $U\cap W$ is an open subset of $U$ which is nonempty, since $\emptyset\ne V\subseteq U\cap W,$ and disjoint from $N,$ since $U\cap W\cap N\subseteq W\cap N=W\cap Y\cap N=V\cap N=\emptyset.$
Therefore, if $M\subseteq Y\subseteq X$ and $M$ is meager in $Y,$ then $M$ is meager in $X.$
Therefore, if $Y$ is comeager in $X$ and $A$ is comeager in $Y,$ then $Y\setminus A,$ being comeager in $Y,$ is also comeager in $X,$ and so is $X\setminus A=(X\setminus Y)\cup(Y\setminus A),$ whence $A$ is comeager in $X.$