  Using three correlated band approaches, namely the conventional band approach
plus on-site Coulomb repulsion $U$, the modified Becke-Johnson functional, and
hybrid functional, we have investigated inverse-Heusler ferrimagnets
Cr$_2$Co${\cal Z}$ (${\cal Z}$=Al, Ga, In). These approaches commonly indicate
that the Cr$_2$CoAl synthesized recently is a precise compensated half-metal
(CHM), whereas Cr$_2$CoGa and Cr$_2$CoIn are ferrimagnets with a small moment.
This is also confirmed by the fixed spin moment approach. Analysis of the Bader
charge decomposition and the radial charge densities indicates that this
contrast is due to chemical differences among the ${\cal Z}$ ions.
Additionally, in Cr$_2$CoAl, changing the volume by $\pm$ 5% or the ratio of
$c/a$ by $\pm$ 2% does not alter the CHM state, suggesting that this state is
robust even under application of moderate pressure or strain. Considering the
observed high Curie temperature of 750 K, our results suggest that Cr$_2$CoAl
is a promising candidate for robust high $T_C$ CHMs. Furthermore, the
electronic structure of the CHM Cr$_2$CoAl is discussed.
