Fixed points under action of inverse flower arrangement form flower arrangement

Statement
Let $$G$$ be a group with a group action on a set $$X$$. Let $$H_i, i \in I$$ be an inverse flower arrangement of subgroups of $$G$$. Let $$A_i$$ be the set of points of $$X$$ fixed by every element of $$H_i$$. Then, the $$A_i$$s form a flower arrangement of subsets of $$X$$.