Let $x$ and $y$ be two elements of the five-dimensional complex vector space $\mathbb{C}^{5}$, and let $A$ be a set, such that each of its elements can be written as $(\alpha x+\beta y)$, with $\alpha$ and $\beta$ two complex numbers. Discuss the existence and properties of a possible subgroup of $\mathrm{U}(5)$ leaving $A$ invariant.