Let $X$ be a simplicial topological space, with a fixed simplicial decomposition whose vertices are denoted by $i,j,k,...$. For any abelian group $A$ we can consider the singular cohomology groups $H^k(X,A)$ whose classes are represented by $k-$cochains assigning an element of $A$ for each $k-$dimensional simplex, namely $a_{i_0,i_1,...,i_k}\in A$, satisfying the cocycle condition (I am using additive notation for abelian groups) $$ (d a)_{i_0,...,i_{k+1}}=\sum _{j=0}^{k+1}(-1)^j a_{i_0,...,\widehat{i_j},...,i_{k+1}}=0 $$ If I have two abelian groups $A_1,A_2$, and a second group cohomology class $\epsilon \in H^2(A_2,A_1)$ I can uniquely construct the central extension $A$ of $A_2$ by $A_1$. This is associated with a short exact sequence $$ 1\rightarrow A_1\rightarrow G\rightarrow A_2\rightarrow 1 $$ Let us assume $G$ to be abelian too. The class $\epsilon$ is an obstruction to have a section $s: A_2\rightarrow G$ which is a group homomorphism.
Now associated with this sequence we have a long exact sequence of cohomology groups of $X$ valued in the three abelian groups, with connected homomorphisms given by the Bockstein homomorphisms $$ \beta : H^k(X,A_2)\rightarrow H^{k+1}(X,A_1) $$ Thus given a class $[a]\in H^k(X,A_2)$ represented by a cocycle $a$, I have a corresponding class $\beta ([a])\in H^{k+1}(X,A_1)$. I am looking for an explicit formula for a representative of this class, in terms of $a$ and the class $\epsilon \in H^2(A_2,A_1)$ which determines the extension.
In the case $k=1$ I have a conjecture: $$ \beta(a)_{i,j,k}=\epsilon(a_{ij}+a_{jk},-a_{ij})+\epsilon(a_{ij},a_{jk}) $$ But I don't know whether this is correct or not, and in case how to prove it.
Moreover, is there any expression for generic $k$?
As was mentioned in the comments, this follows from unraveling the definitions. So let $0 \to A_1 \to G \to A_2 \to 0$ be a group extension with $G$ abelian. Fix a section $s\colon A_2\to G$ of the projection with $s(0) = 0$. We obtain a $2$-cocycle $\epsilon\colon A_2\times A_2 \to A_1$ via $$ s(a) + s(b) = s(a+b) + \epsilon(a,b). $$ Note that $\epsilon(a,b) = 0$ if either $a=0$ or $b=0$, and that $$ s(a) - s(b) = s(a-b) - \epsilon(a-b,b). $$ Now, given a $k$-cocycle $c\colon X_k \to A_2$ (where $X_k$ denotes the set of $k$-simplices in the fixed simplicial decomposition of $X$), we obtain a $k+1$-cocycle $$ (sc)\circ \partial \colon X_{k+1} \to G, $$ where $sc$ is the composite $X_k\xrightarrow{c} A_2\xrightarrow{s} G$. It is easy to check that $(sc)\circ \partial$ takes values in $A_1$, and so the Bockstein homomorphism is given by $$ \beta\colon H^k(X,A_2) \longrightarrow H^{k+1}(X,A_1),\qquad [c] \longmapsto [(sc)\circ \partial]. $$ To avoid clunky notation, I will write $c_{i_j} := c(i_0,\dotsc, i_{j-1},i_{j+1},\dotsc,i_k)$ and $c_l := 0$ if $l>k+1$ in the following: we compute \begin{align*} \bigl((sc)\circ &\partial)(i_0,\dotsc,i_{k+1}) = \sum_{j=0}^{k+1} (-1)^j sc_{i_j}\\ &= s(c_{i_0}-c_{i_1}) - \epsilon(c_{i_0}-c_{i_1},c_{i_1}) + \sum_{j=2}^{k+1} (-1)^j sc_{i_j}\\ &= s(c_{i_0} - c_{i_1} + c_{i_2}) - \epsilon(c_{i_0}-c_{i_1},c_{i_1}) + \epsilon(c_{i_0}-c_{i_1}, c_{i_2}) + \sum_{j=3}^{k+1} (-1)^jsc_{i_j} \\ &= \dotsb = \underbrace{s\Bigl(\sum_{j=0}^{k+1}(-1)^jc_{i_j}\Bigr)}_{=0} + \sum_{j=0}^{\lfloor k/2\rfloor} \left( -\epsilon\Bigl(\sum_{l=0}^j (c_{2l} - c_{2l+1}), c_{2j+1}\Bigr) + \epsilon \Bigl(\sum_{l=0}^j (c_{2l}- c_{2l+1}), c_{2j+2}\Bigr)\right), \end{align*} where the first summand vanishes, because $c$ is a cocycle and $s(0) = 0$. This gives the concrete description of $\beta$ in terms of $\epsilon$.
Specifically for $k=0$, we have $$ \beta(c)(i_0,i_1) = -\epsilon\bigl(c(i_1) - c(i_0),c(i_1)\bigr), $$ and for $k=1$, we obtain $$ \beta(c)(i_0,i_1,i_2) = -\epsilon\bigl(c(i_1,i_2) -c(i_0,i_2), c(i_0,i_2)\bigr) + \epsilon\bigl(c(i_1,i_2) - c(i_0,i_2), c(i_0,i_1)\bigr). $$