  We study a new class of square Sierpi\'nski carpets $F_{n,p}$ ($5\leq n,
1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically
equivalent to the standard Sierpi\'{n}ski carpets. We prove that the group of
quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group.
  We also establish that $F_{n,p}$ and $F_{n',p'}$ are quasisymmetrically
equivalent if and only if $(n,p)=(n',p')$.
