On weak$^*$-extensible subspaces of Banach spaces

Let $X$ be a Banach space and $Y \subseteq X$ be a closed subspace. We prove that if the quotient $X/Y$ is weakly Lindel\"{o}f determined or weak Asplund, then for every $w^*$-convergent sequence $(y_n^*)_{n\in \mathbb N}$ in $Y^*$ there exist a subsequence $(y_{n_k}^*)_{k\in \mathbb N}$ and a $w^*$-convergent sequence $(x_k^*)_{k\in \mathbb N}$ in $X^*$ such that $x_k^*|_Y=y_{n_k}^*$ for all $k\in \mathbb N$. As an application we obtain that $Y$ is Grothendieck whenever $X$ is Grothendieck and $X/Y$ is reflexive, which answers a question raised by Gonz\'{a}lez and Kania.