Given an extension of groups
$$ 1 \to H \to G \to Q \to 1,$$
there is a spectral sequence 
$$E^{ip}\_2(M) = H^i(Q,H^p(H,M)) \Rightarrow H^{i+p}(G,M).$$
I understand that the composition of the cup products for $Q$ and $H$ defines a pairing 
$$ E\_2^{ip}(M) \otimes E\_2^{jq}(N)\hspace{180pt}$$
$$\begin{array}{cl}
= & H^i(Q,H^p(H,M) \otimes H^j(Q,H^q(H,M) \newline 
\xrightarrow[]{\cup\_Q} & H^{i+j}(Q,H^p(H,M) \otimes H^q(H,N)) \newline 
\xrightarrow[]{\cup\_H^\ast} & H^{i+j}(Q,H^{p+q}(H,M \otimes N)) \newline 
= & E\_2^{i+j,p+q}(M \otimes N).
\end{array}$$
But according to section 7.3 in L. Evens' book (The Cohomology of Groups), the sign 
$$(-1)^{pj}$$
is needed in order to make this pairing a proper product in the spectral sequence. 


*Question:* Where does this sign come from ? 



Sign in the product of the LHS spectral sequence 
Let $X \to k$ resp. $Y \to k$ be projective resolutions of $k$ over $kG$ resp. $kQ$. In short, the reason for the sign is the twist 
$$T : X \otimes Y \to Y \otimes X,\; x \otimes y \mapsto (-1)^{ij} \cdot y \otimes x\quad,\quad x \in X\_i, y \in Y\_j.$$


In detail: First note that if $U \to k$ is a projective resolution of $k$ over $kG$ and $\Delta: U \to U \otimes U$ is a diagonal approximation, then (using Evens' sign conventions) cup-product $$Hom\_G(U,M) \otimes Hom\_G(U,N) \to Hom\_G(U,M\otimes N)$$ is given as follows: Let $\Delta\_{ij}: U\_{i+j} \to U\_i \otimes U\_j$ be the $(i,j)$-component of $\Delta$. If $\Delta\_{ij}(u) = u\_i \otimes u\_j$ then 
$$(f \cup g)(u) = f(u\_i) \otimes g(u\_j).\hspace{80pt}(\ast)$$
Now let $\Delta\_X: X \to X \otimes X$ and $\Delta\_Y: Y \to Y \otimes Y$ be diagonal approximations. $X \otimes Y \to k$ is a projective resolution for $G$ (diagonal operation) and 
$$\Delta: X \otimes Y \xrightarrow{\Delta\_X \otimes \Delta\_Y} X \otimes X \otimes Y \otimes Y 
\xrightarrow{\;T\;} (X\otimes Y) \otimes (X \otimes Y)$$
is a diagonal approximation. 


Let $x \in X\_{i+j}$ and $y \in Y\_{p+q}$. If $\Delta\_{i,j}(x) = x\_i \otimes x\_j$ and $\Delta\_{p,q}(y) = y\_p \otimes y\_q$ then 
$$\Delta\_{i+p,j+q}(x \otimes y) = T(x\_i \otimes x\_j \otimes y\_p \otimes y\_q) =(-1)^{jp}(x\_i \otimes y\_p) \otimes (x\_j \otimes y\_q).$$ 


Hence we have for the cup-product 
$$\cup: Hom\_G(X \otimes Y,M) \otimes Hom\_G(X \otimes Y,N) \to Hom\_G(X \otimes Y,M \otimes N)$$
 $$(f \cup g) (x \otimes y) = (-1)^{jp} f(x\_i \otimes y\_p) \otimes g(x\_j \otimes y\_q).\quad\quad(1)$$


One can check that this product preserves the usual filtration and hence defines a product on the spectral sequence that is compatible with the product in the cohomology of $G$. 


Since your pairing is a composition of the cup-product on $Q$ and on $H$, it can be seen from $(\ast)$ that no sign occurs there. Furthermore, using the isomorphism 
$$Hom\_G(X \otimes Y,-) \cong Hom\_Q(Y,Hom\_H(X,-))$$ 
of complexes, one obtains by definition checking - use $(\ast)$ - that your pairing is given on cochain level by 
$$Hom\_G(X \otimes Y,M) \otimes Hom\_G(X \otimes Y,N) \to Hom\_G(X \otimes Y,M \otimes N)$$
$$(f \cup g) (x \otimes y) = f(x\_i \otimes y\_p) \otimes g(x\_j \otimes y\_q).$$ 
This shows that the two products differ by the sign $(-1)^{jp}$. 


**Remark:** The sign depends on the definition of the differential in the $Hom$-cocomplex (that influences $(\ast)$). See for example the paper [Hochschild, Serre: Cohomology of Group Extensions, Trans. Amer. Math. Soc. 74(1),1953, pp. 110-134] Chap. II, Theorem 3 for a different sign. 


**Added:** To explicate this remark, consider the sign convention in Brown's group cohomology book. He uses $$\delta: Hom\_G(U\_n,M) \to Hom\_G(U\_{n+1},M),\; f \mapsto (-1)^{n+1}f \circ d\_{n+1}.$$ This leads to the cup product 
$$(f \cup g)(u) = (-1)^{ij} \cdot f(u\_i) \otimes g(u\_j)\hspace{80pt}(\ast')$$ 
Thus we obtain in place of $(1)$: 
$$(f \cup g)(x \otimes y) = (-1)^{jp} (-1)^{(i+p)(j+q)} f(x\_i \otimes y\_p) \otimes g(x\_j \otimes y\_q)\quad\quad (1')$$
Let $F \in Hom\_H(Y\_p,Hom\_Q(X\_i,M))$ correspond to $f \in Hom\_G( (X\otimes Y)\_{i+p},M)$ and $G \in Hom\_H(Y\_q,Hom\_Q(X\_j,M))$ correspond to $g \in Hom\_G( (X\otimes Y)\_{j+q},N)$. 
Then by $(\ast')$:
$$(F \cup\_Q G)(y) = (-1)^{pq} \cdot F(y\_p) \otimes F(y\_q).$$
Thus we obtain for your pairing 
$$(F \cup G)(y)(x) = (-1)^{pq} \big (F(y\_p) \cup\_H F(y\_q) \big )(x)
= (-1)^{pq}(-1)^{ij} F(y\_p)(x\_i) \otimes F(y\_q)(x\_j).$$
Comparing with $(1')$ shows that the two products differ by the sign $(-1)^{iq}$, in accordance with Hochschild-Serre. 


