Let X be a $T\sb{2}$ Baire space. A set $B \subset {X}$ has the Baire property if there exists an open set $U \subset {X}$ such that ($U\\{B}) \cup (B\\{U})$ is meager. A system, S, of subsets of X is strongly transitive if for any A $\in$ S having the Baire property, either A or $X\\{A}$ is meager. In 1932, Kuratowski proved that the system consisting of all unions of composants of the simplest Knaster continuum, $B\sb{0}$, is strongly transitive. He then asked whether this were the case for every indecomposable continuum. Herein, an attempt is made to answer this question, although unsuccessfully, via a mapping approach. Principal result: If X is a metric indecomposable continuum which admits a finite-to-one map onto $B\sb{0}$ then the system of all unions of composants of X is strongly transitive. More important than this however, is the illumination of the structure of strongly transitive systems in Chapter III: several notions are generalized, a convenient framework for their study is provided and some related concepts are introduced.