For discrete groups, we construct two bounded cohomology classes with coefficients in the second space of the reduced real $\ell_1$-homology. Precisely, we associate to any discrete group $G$ a bounded cohomology class of degree two noted $\frak{ g}_2\in H_b^2(G, \bar{ H}_2^{ \ell_1}(G, \mathbb R))$. For $G$ and $\Pi$ groups and $\theta : \Pi\rightarrow Out(G)$ any homomorphism we associate a bounded cohomology class of degree three noted $[\theta]\in H_b^3(\Pi, \bar{ H}_2^{ \ell_1}(G, \mathbb R))$. When the outer homomorphism $\theta : \Pi\rightarrow Out(G)$ induces an extension of $G$ by $\Pi$ we show that the class $\frak{ g}_2$ is $\Pi$-invariant and that the differential $d_3$ of Hochschild-Serre spectral sequence sends the class $\frak{ g}_2$ on the class $[\theta]$ : $d_3(\frak{ g}_2)=[\theta]$. Moreover, we show that for any integer $n\geq 0$ the differential $d_3 : E_3^{ n, 2}\rightarrow E_3^{ n+3, 0}$ of Hochschild-Serre spectral sequence in real bounded cohomology is given as a cup-product by the class $[\theta]$.
Categories: math.AT