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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$\oplus_\infty^n\mathcal{D}(\mathbb{Z})\hookrightarrowH_b^2(\Gamma,H;\mathbb{R})$, where $\mathcal{D}(\mathbb{Z})$ is the space ofbounded alternating functions on $\mathbb{Z}$ equipped with the defect norm.

Relative second bounded cohomology of free groups