Consider a finite dimensional $C^*$-algebra $\cal{A}$. Is there any enveloping $C^*$-algebra $\cal{C^*(G)}$ such that $\cal{A}\cong C^*(G)$ for some locally compact group $\cal{G}$?
(Note that "$\cong$" is the $C^*$-algebra isomorphism.
Consider a finite dimensional $C^*$-algebra $\cal{A}$. Is there any enveloping $C^*$-algebra $\cal{C^*(G)}$ such that $\cal{A}\cong C^*(G)$ for some locally compact group $\cal{G}$?
(Note that "$\cong$" is the $C^*$-algebra isomorphism.
No, in general.
If $C^\ast(G)$ is to be finite-dimensional then $G$ must be a finite group, and so $C^*(G)= \mathbb{C}(G)$ is complex group algebra of $G$. Basic results from the representation theory of finite groups identify this algebra, up to $*$-isomorphism, as follows:
$$ C^*(G) \cong \bigoplus_{[\pi]\in \widehat{G}} M_{d_\pi}(\mathbb C),$$
where $\widehat{G}$ is the set of isomorphism classes of irreducible representations of $G$, and $d_\pi$ is the dimension of the vector space underlying the representation $\pi$. Since every group admits a one-dimensional representation (the trivial representation), $C^*(G)$ will always have a one-dimensional summand.
The problem of characterising the group algebras of finite groups among all finite-dimensional $C^*$-algebras---that is, the problem of characterising the multisets $\{ d_\pi \ |\ [\pi]\in \widehat{G}\}$ among all finite multisets of positive integers---is still open; see here for instance.