 Solution

Prepare an ancillary qubit inIt can be prepared from $|0⟩$ by an $X$ operation followed by a Hadamard gate, independently of $g$. One call to the Boolean quantum oracle then gives quantum phase kickback:The data state receives a minus sign precisely on the basis vectors spanning $\mathcal{G}$, while the ancilla remains unchanged and unentangled. One oracle call realizes the reflection. By linearity it is the marked-state phase oraclewith one oracle call. The ancillary preparation and its optional inverse use only fixed operations, so no additional knowledge of $g$ is needed.