 First homotopy group above the dimension of a sphere

The universal cover of $S^1$ gives the first case. The Hopf fibration $S^1\to S^3\to S^2$ identifies ${\pi }_3(S^2)$ with ${\pi }_3(S^3)=\mathbb{Z}$. Starting from ${\pi }_4(S^3)=\mathbb{Z}/2$, the Freudenthal suspension theorem makes every subsequent suspension in these degrees an isomorphism. The stable range does not include the first step from ${\pi }_3(S^2)$ to ${\pi }_4(S^3)$, so the integer group at $n=2$ is not a contradiction.