Topological groups and Dugundji compacta

A compact space $X$ is called a Dugundji compactum if for every compact $Y$ containing $X$, there exists a linear extension operator
$$\Lambda\colon C(X)\to C(Y),$$
which preserves nonnegativity and maps constants into constants. It is known that every compact group is a Dugundji compactum. In this paper we show that compacta connected in a natural way with topological groups enjoy the same property. For example, in each of the following cases, the compact space $X$ is a Dugundji compactum:
1) $X$ is a retract of an arbitrary topological group;
2) $X=\beta P$, where $P$ is a pseudocompact space on which some $\aleph_0$-bounded topological group acts transitively and continuously.
Bibliography: 57 titles.