Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?

Question

A topological space $X$ is called a $\sigma$-space if every $F_{\sigma}$-subset of $X$ is $G_{\delta}$.

A topological space $X$ is called a $Q$-space if any subset of $X$ is $F_{\sigma}$.

Definition. A topological space $X$ is called a hereditary $\sigma$-space if every subset of $X$ is $\sigma$-space.

Question. Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space ?

Answer 1

Every $S_1(B_\Gamma,B_\Gamma)$ space is a $\sigma$-space, and the property $S_1(B_\Gamma,B_\Gamma)$ is hereditary for subsets (B. Tsaban and M. Scheepers, The combinatorics of Borel covers, Topology and its Applications 121 (2002), 357-382.)

For example, a Sierpiński set satisfies $S_1(B_\Gamma,B_\Gamma)$.

On the other hand, it is very difficult to be a Q-set. For example, Q-sets have Lebesgue measure zero. In particular, they cannot be Sierpiński.