Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and $\mathrm{char}K\ne p$. Suppose that a classical group $G$ acts on the $F$-vector space $V$. Then it can induce the actions on the vector space $V\oplus V$ and on the group algebra $K[V\oplus V]$, respectively. In this paper we determine the structure of $G$-invariant ideals of the group algebra $K[V\oplus V]$, and establish the relationship between the invariant ideals of $K\left[V\right]$ and the vector invariant ideals of $K[V\oplus V]$, if $G$ is a unitary group or orthogonal group....