Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\sqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=spec(k)$, then $S^\bullet(X)$ is the universal monoidal $k$-scheme. Now if $G$ is a commutative quasi-projective $k$-group, then $S^\bullet(G)$ is defined and, moreover, there is a map of algebraic groups $S^d(G)\to G$ given by $d$-fold summation map $G^d\to G$.

If $\mathcal{C}$ is the category of quasi-protective $k$-varieties and $\mathcal{D}$ is the category of monoidal quasi-projective $k$-varieties, then $S^\bullet$ is left adjoint to a forgetful functor $U$. This defines a comonad on $\mathcal{D}$ by taking $\bot=(S^\bullet\circ U)\colon\mathcal{D}\to \mathcal{D}$. Thus one has an augmented simplicial monoidal (commutative) $k$-scheme $\bot_*(G)\to G$ such that $U(\bot_*(G))\to U(G)$ is contractible (Proposition 8.6.10 of Weibel's book). This gives extra degeneracies for simplicial scheme $S^\bullet_*(G)\to G$, its right part is given by $S^\bullet(S^\bullet(G))\rightrightarrows S^\bullet(G)\to G$. Here by $S^\bullet_n(G)$ I mean applying $S^\bullet(-)$ to $G$ iteratively $n$ times.

If $V$ is any smooth $k$-variety, I could apply $Mor_{Sch_k}(V, -)$ to this simplicial scheme and get an augmented simplicial set $Mor(V, S^\bullet_*(G))\to Mor(V, G)$. As far as I understand the previous shows that this is a contractible simplicial set but all the maps (except of extra degeneracies) are also the maps of monoids. A priori, extra degeneracies are only maps of sets, but I guess they also should be maps of monoids. Is there any simple way to see this?

Why do I care about this? If $char(k)=0$ then the groupification of each commutative monoid $Mor(V, S^\bullet_*(G))$ is the group of correspondences $Cor(V, S^\bullet_{*-1}(G))$. Thus if the above simplicial guy is contractible (as commutative monoid), I think that I could take the gropification and get contractible augmented simplicial abelian group. After applying Dold-Kan we see that the complex $$\ldots\to Cor(V, S^\bullet(S^\bullet(G)))\to Cor(V, S^\bullet(G))\to Cor(V, G)\to Mor(V, G)$$ is exact and thus we can conclude that there is an exact sequence of presheaves with transfers given by $$\ldots\to\mathbb{Z}_{tr}(S^\bullet(S^\bullet(G)))\to \mathbb{Z}_{tr}(S^\bullet(G))\to\mathbb{Z}_{tr}(G)\to G^{tr}.$$

Does anyone has any suggestion how to modify the above construction or how to make it clearer? Anything could be helpful.

Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then $S^\bullet(X)$ is the universal monoidal $k$-scheme. Now if $G$ is a commutative quasi-projective $k$-group, then $S^\bullet(G)$ is defined and, moreover, there is a map of algebraic groups $S^d(G)\to G$ given by $d$-fold summation map $G^d\to G$.

If $\mathcal{C}$ is the category of quasi-protective $k$-varieties and $\mathcal{D}$ is the category of monoidal quasi-projective $k$-varieties, then $S^\bullet$ is left adjoint to a forgetful functor $U$. This defines a comonad on $\mathcal{D}$ by taking $\bot=(S^\bullet\circ U)\colon\mathcal{D}\to \mathcal{D}$. Thus one has an augmented simplicial monoidal (commutative) $k$-scheme $\bot_*(G)\to G$ such that $U(\bot_*(G))\to U(G)$ is contractible (Proposition 8.6.10 of Weibel's book). This gives extra degeneracies for simplicial scheme $S^\bullet_*(G)\to G$, its right part is given by $S^\bullet(S^\bullet(G))\rightrightarrows S^\bullet(G)\to G$. Here by $S^\bullet_n(G)$ I mean applying $S^\bullet(-)$ to $G$ iteratively $n$ times.

If $V$ is any smooth $k$-variety, I could apply $\operatorname{Mor}_{\operatorname{Sch}_k}(V, -)$ to this simplicial scheme and get an augmented simplicial set $\operatorname{Mor}(V, S^\bullet_*(G))\to \operatorname{Mor}(V, G)$. As far as I understand the previous shows that this is a contractible simplicial set but all the maps (except of extra degeneracies) are also the maps of monoids. A priori, extra degeneracies are only maps of sets, but I guess they also should be maps of monoids. Is there any simple way to see this?

Why do I care about this? If $\operatorname{char}(k)=0$ then the groupification of each commutative monoid $\operatorname{Mor}(V, S^\bullet_*(G))$ is the group of correspondences $\operatorname{Cor}(V, S^\bullet_{*-1}(G))$. Thus if the above simplicial guy is contractible (as commutative monoid), I think that I could take the gropification and get contractible augmented simplicial abelian group. After applying Dold-Kan we see that the complex $$\ldots\to \operatorname{Cor}(V, S^\bullet (S^\bullet(G))) \to \operatorname{Cor}(V, S^\bullet(G))\to \operatorname{Cor}(V, G)\to \operatorname{Mor}(V, G)$$ is exact and thus we can conclude that there is an exact sequence of presheaves with transfers given by $$\ldots\to\mathbb{Z}_{\operatorname{tr}}(S^\bullet(S^\bullet(G)))\to \mathbb{Z}_{\operatorname{tr}}(S^\bullet(G))\to\mathbb{Z}_{\operatorname{tr}}(G)\to G^{\operatorname{tr}}.$$

Does anyone has any suggestion how to modify the above construction or how to make it clearer? Anything could be helpful.