The long exact sequence associated to the Hochschild-Serre spectral sequence for extension of groups $1 \to H \to G \to G/H \to 1$ is $$ \begin{array}[t]{lll} 1 \to & H^1(G/H, A^{G/H}) \xrightarrow{\inf} H^1(G,A) \xrightarrow{\mathrm{res}} H^1(H, A)^{G/H} \xrightarrow{\mathrm{tr}} & \\[1ex] & \to H^2(G/H,A^H) \xrightarrow{\inf} \mathrm{Ker}(H^2(G, A) \to H^2(H,A)) \xrightarrow{\rho} \\[1ex] & \xrightarrow{\rho} H^1(G/H, H^1(H, A)) \xrightarrow{\mathrm{tr}} H^3(G/H, A^H) &\\ \end{array} $$

I am interested in the morphism $\rho$ described explicitly in terms of cocycles (as in the standard elementary definition of Galois cohomology; these are called "inhomogeneous" cocycles in the book of Neukirch, Schmidt and Wingberg). I realize that this is just an exercise consisting of writing down the spectral sequence of the double complex and unwinding the definitions, but I wonder if this has already been done somewhere in an easily quotable form.