Relative second bounded cohomology of free groups

This paper is devoted to the computation of the space $H_b^2(\Gamma,H;\mathbb{R})$, where $\Gamma$ is a free group of finite rank $n\geq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in $\Gamma$ if and only if $H_b^2(\Gamma,H;\mathbb{R})$ is not trivial, and furthermore, if and only if there is an isometric embedding $\oplus_\infty^n\mathcal{D}(\mathbb{Z})\hookrightarrow H_b^2(\Gamma,H;\mathbb{R})$, where $\mathcal{D}(\mathbb{Z})$ is the space of bounded alternating functions on $\mathbb{Z}$ equipped with the defect norm.