I was reading about cohomology and long exact sequences. I found that
Given $$0 \to L \to M \to N \to 0$$ is a short exact sequence of $G$- modules, then a there exists a long exact sequence is defined as $$0\longrightarrow L^G \longrightarrow M^G \longrightarrow N^G \overset{\delta^0}{\longrightarrow} H^1(G,L) \longrightarrow H^1(G,M) \longrightarrow H^1(G,N) \overset{\delta^1}{\longrightarrow} H^2(G,L)\longrightarrow \cdots$$ and the so-called connecting homomorphism is: $$\delta^n : H^n (G,N) \to H^{n+1}(G, L)$$
My specific question is for $n=1$ :
Is there any explicit description of the map $$\delta^1 : H^1 (G,N) \to H^{2}(G, L)$$
I have found here that:
The description of the map $\delta^n$ is given as:
If $c \in H^n(G, N)$ is represented by an $n$-cocycle $\phi: G^n \to N$ then $\delta^n(c)$ is $d^n(\varphi)$ where $\varphi$ is an $n$-cochain $G^n \to M$ "lifting" $\phi$ via surjective map $M\to N$ and $d^n$ is defined (here) as $$\left(d^{n+1}\varphi\right) (g_1, \ldots, g_{n+1}) = g_1 \varphi(g_2, \dots, g_{n+1}) + \sum_{i=1}^n (-1)^i \varphi \left (g_1,\ldots, g_{i-1}, g_i g_{i+1}, \ldots, g_{n+1} \right ) + (-1)^{n+1}\varphi(g_1,\ldots, g_n)$$
But from the above description is it seems that $d^n(\varphi) : G^{n+1}\to M$ but we need that $\delta^n(c) : G^{n+1}\to L$, so what I am actually missing here?
Any help will be greatly appreciated.