  Let $X$ be a finitistic space with its rational cohomology isomorphic to that
of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We
study continuous $\mathbb{S}^1$ actions on $X$ and determine the possible fixed
point sets up to rational cohomology depending on whether or not $X$ is totally
non-homologous to zero in $X_{\mathbb{S}^1}$ in the Borel fibration
$X\hookrightarrow X_{\mathbb{S}^1} \longrightarrow B_{\mathbb{S}^1}$. We also
give examples realizing the possible cases.
