Forgive me if the following questions on homotopy groups are trivial - I am not knowledgeable in the subject.
Is there a manifold $M$ with nontrivial second homotopy group $\pi_2(M)$ but trivial third homotopy group $\pi_3(M)$?
Is there such an example with $M =G/H$ with $G$ a Lie group and $H$ a subgroup of $G$?
The answer is yes to both. As explained on the topospaces wiki, using the long exact sequence in homotopy associated to the fibration $S^1 \to S^{2n+1} \to \mathbb{CP}^n$ you can compute that $\pi_2(\mathbb{CP}^n) \cong \mathbb{Z}$ but that $\pi_k(\mathbb{CP}^n) = 0$ for $2 < k < 2n+1$. So $\mathbb{CP}^n$ is an example for all $n \ge 2$, and it can also be written as a quotient $U(n+1)/(U(n) \times U(1))$.