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.



the sixth morphism in the long exact sequence associated to the Hochschild-Serre spectral sequence
Yes, the morhpisms have been described in detail in the paper "A seven-term exact sequence for the cohomology of a group extension" by Dekimpe, Hartl and Wauters: see <http://arxiv.org/pdf/1103.4052.pdf>.   

Section $6$ gives details for $\rho$ (the sequence is slightly different, but applies more or less to your situation). This morphism is induced by differentials in the spectral sequence, and so far there was no explicit description available, but in this paper a purely group-theoretical description is given.


