Let $F$ be a free group. Let $F_1$ be a subgroup with basis $B$. Assume that $F_1$ has the property that for every $\alpha\in F-F_1$, the set $\{\alpha\}\cup B$ is still free.
Is there a name for such subgroup $F_1$ with this property? Or are there known characterizations of these subgroups? We might restrict both the rank of $F$ and $F_1$ to be finite.