Homoflatness on Ideal Extensions

We consider monoids $S=G\dot{\cup} I$ where $G$ is a group and $I$ is an ideal of $S$ and show that if an $S$-act is principally weakly homoflat or weakly homoflat as an $I^1$-act, then it has these properties as an $S$-act. We also show that an $S$-act which is (weakly) pullback flat, equalizer flat, (principally) weakly kernel flat, translation kernel flat or satisfies Condition $(E)$ as an $I^1$-act may not generally have these properties as an $S$-act. The flatness notions considered in this paper were introduced in {\it V. Laan, Pullbacks and flatness properties of acts I, Comm. Alg. ${\bf 29}(2)$ (2001), 829--850}.