Are there two sterile neutrinos cooscillating with $\nu_e$ and $\nu_\mu$?

The existence of two sterile neutrinos $\nu_s$ and $\nu'_s$ (blind to all Standard-Model interactions) is shown to be implied by a model of fermion "texture" that we develop since some time. They may mix nearly maximally with two of three conventional neutrinos, say $\nu_e$ and $\nu_\mu$, thus leading to neutrino oscillations, say $\nu_e \to \nu_s$ and $\nu_\mu \to \nu'_s$, with nearly maximal amplitudes. Then, they can be responsible for the observed deficits of solar $\nu_e$'s and atmospheric $\nu_\mu$'s, respectiv% ely, but by themselves do not help to explain the LSND results for $\nu_\mu \to \nu_e$ oscillations. On the other hand, they are consistent with the CHOOZ negative result. At the moment, the experiment cannot decide, whether the deficit of atmospheric $\nu_\mu$'s, confirmed by the recent \SK findings, has to be related to the oscillations $\nu_\mu \to \nu_\tau$ or $\nu_\mu \to \nu'_s$. In the last Section of the paper, a new notion of "non-Abelian spin-1/2 fermions" is presented in the context of a composite option for fermion families.