Let \(X = G/H\) be a homogeneous space, with \(G\) a connected Lie group, \(H\) a closed connected subgroup. The author examines the question whether and when there are closed connected subgroups \(L\) of \(G\) acting properly on \(X\) with compact quotient \(L \backslash X\). In view of sufficient criteria for this, he also undertakes to find conditions under which the compactness of every stabilizer \(L_x\) for \(x \in X\) (``condition CI'') implies properness of the action of \(L\). These questions have been posed by \textit{T. Kobayashi} [Discontinuous groups acting on homogeneous spaces of reductive type, in `Representation Theory of Lie groups and Lie Algebras', Proc. Conf. Fuji-Kawaguchiko/Japan 1990, 59-75 (1992)]. The present paper contains answers to these questions in specific situations. The existence of a subgroup \(L\) acting properly and with compact quotient is first given the obvious affirmative answer in the case of a semidirect product \(G = H \times V\) where \(V\) is a normal vector group (take \(L = V)\). Then, the case that \(G\) is a simply connected nilpotent Lie group is studied. It is shown that a nontrivial closed connected subgroup \(L\) acting properly on \(X\) always exists, and a necessary and sufficient condition for the existence of such a subgroup with compact quotient is established in terms of the Lie algebras involved. The second question is answered for certain semidirect products \(G = H \times V\), namely for \(H = \text{PSL} (2, \mathbb{R} )\) and an arbitrary normal vector group \(V\), and for \(H = N_3 (\mathbb{R})\), the group of upper triangular \(3 \times 3\) real matrices, in its natural action on \(V = \mathbb{R}^3\). It is shown that then condition CI for a closed connected subgroup \(L\) of \(G\) implies that \(L\) acts properly on \(X\). The analogous case \(H = N_2 (\mathbb{R})\), \(V = \mathbb{R}^2\) has been dealt with by Kobayashi [loc. cit.]. The last section presents the following reduction of this problem: if one could prove the above implication \((\text{CI} \Rightarrow\) properness) for unipotent closed connected subgroups \(L\) of the semidirect products \(N_r (\mathbb{R}) \times \mathbb{R}^r\) for arbitrary dimension \(r\), where \(N_r (\mathbb{R})\) is the group of upper triangular \(r \times r\) real matrices, then the same implication would hold more generally for any algebraic subgroup \(L\) of a semidirect product \(G = H \times V\) where \(G\) is algebraic, \(V\) a normal vector subgroup, and \(H\) reductive and effective on \(V\). The proof of this reduction is based on a theorem on structural implications of the condition CI for a connected algebraic subgroup \(L\) of \(G\) in the case that \(G\) is algebraic and \(H\) an algebraic subgroup containing a Levi factor of \(G\). The latter theorem is also of independent interest; in fact it is a strengthening in the algebraic situation of a theorem by Kobayashi [loc. cit.].