Let $\mathcal O$ be a principal ideal domain with field of fractions $F$. Let $G = \operatorname{Spec} A$ be a linear algebraic group over $F$, with comultiplication map $d: A \rightarrow A \otimes_F A$. In other words, $G$ is a group scheme over $F$, and $A$ is a finite type and geometrically reduced $F$-algebra.
In the article $\mathfrak p$-adic Groups by F. Bruhat, he defines an $\mathcal O$-structure on $G$ to be an $\mathcal O$-subalgebra $A_0$ of $A$ such that $A = FA_0$ and $d(A_0) \subset A_0 \otimes_{\mathcal O} A_0$.
Since $A_0$ is torsion free and hence flat over $\mathcal O$, we always have $A_0 \subset A_0 \otimes_{\mathcal O} F \subset A$ and $A_0 \otimes_{\mathcal O} A_0 \subset A \otimes_F A$. If we set $G_0 = \operatorname{Spec} A_0$, then the first condition $A = FA_0$ is equivalent to saying that $$G = G_0 \times_{\mathcal O} \operatorname{Spec}F \tag{1}$$ and the second condition $d(A_0) \subset A_0 \otimes_{\mathcal O} A_0$ is equivalent to saying that the multiplication map $\mu: G \times_F G \rightarrow G$ arises from a (necessarily unique) morphism of $\mathcal O$-schemes
$$\mu_0: G_0 \times_{\mathcal O} G_0 \rightarrow G_0 \tag{2}$$ It seems clear to me that $G_0$ is a group monoid over $\mathcal O$, but not necessarily a group scheme. However, as Bruhat claims later, every $\mathcal O$-structure on $G$ is a group scheme over $\mathcal O$ ($\S$ 1, Example).
If $A_0$ is an $\mathcal O$-structure for $G$, is $\operatorname{Spec} A_0$ necessarily a group scheme? It doesn't seem like this should be the case, since nothing about group inversion was mentioned.