  Given the symplectic polar space of type $W(5,2)$, let us call a set of five
Fano planes sharing pairwise a single point a Fano pentad. Once 63 points of
$W(5,2)$ are appropriately labeled by 63 non-trivial three-qubit observables,
any such Fano pentad gives rise to a quantum contextual set known as Mermin
pentagram. Here, it is shown that a Fano pentad also hosts another, closely
related contextual set, which features 25 observables and 30 three-element
contexts. Out of 25 observables, ten are such that each of them is on six
contexts, while each of the remaining 15 observables belongs to two contexts
only. Making use of the recent classification of Mermin pentagrams (Saniga et
al., Symmetry 12 (2020) 534), it was found that 12,096 such contextual sets
comprise 47 distinct types, falling into eight families according to the number
($3, 5, 7, \ldots, 17$) of negative contexts.
