Sorry for my bad English.
Let $G$ be an abstract group (if necessary finite), and $k$ be an algebraically closed field.
Now is there a group scheme $X$ over $k$ such that group of $k$-valued point $X(k)=\operatorname {Hom}_k(\operatorname {Spec} k, X)$ is isomorphic to $G$?
Especially when $G$ is finite, is there such X as reduced?
(I mean "scheme over $k$" as separated finite type over $k$.)
Question: "Can we construct reduced group scheme which is same group structure given an abstract group?"
Answer: As mentioned in the comments: You may check explicitly in simple examples that if $G$ is a finite group and $Γ(G)$ the constant group scheme of $G$ (over a field $k$) it follows the $k$-rational points $Γ(G)(k)≅G$ recover $G$. As an example try $\mathbb{Z}/(2)$. The constant group scheme has $A:=k\{e_g:g∈G\}$ as algebra (direct sum of copies of $k$, one for each $g∈G$ and $Δ:A→A⊗_k A$ defined by $Δ(e_g):=∑_{g=στ} e_σ⊗e_τ$. It follows there is an isomorphism $\Gamma(G)(k):=Spec(A)(k)≅G$ as groups. The group scheme $\Gamma(G)$ is reduced: The ring $A$ is a direct product of fields and is reduced.
Note: A group scheme in this sense is a representable functor
$$h_{\Gamma(G)}: \underline{k-alg} \rightarrow \underline{Groups}$$
meaning the induced functor $h_{\Gamma(G)}: \underline{k-alg} \rightarrow \underline{Sets}$ is representable. Its $k$-rational points $\Gamma(G)(k):=Hom_{k-alg}(A,k)$ is therefore canonically a (finite) group, isomorphic to the original group $G$.
Note: The category $\underline{k-alg}$ is not a small category. You may also consider the affine scheme $\Gamma(G):=Spec(A)$. This scheme has a multiplication operation $m$ and an inversion $i$, and the triple $(\Gamma(G), m,i)$ is a group scheme in the "naive" sense. Here "naive" means you work with $((\Gamma(G), \mathcal{O}_{\Gamma(G)}), m,i)$ as a locally ringed space with two morphisms $m,i$, instead of working with the associated functor of points $h_{\Gamma(G)}$.