The structure of the ${\text{Ca}}_{3}{\text{Co}}_{2}{\mathrm{O}}_{6}$ compound is made of chains on a triangular array. The intra- and interchain coupling are ferromagnetic and antiferromagnetic, respectively, and the chain itself contains a 1:1 alternation of ${\text{CoO}}_{6}$ trigonal prisms (TP) and octahedra (oct.). The ${\text{Co}}^{3+}$ spin states are high spin $(S=2)$ and low spin $(S=0)$ in the TP and the oct., respectively. It is found that a few percent of ${\text{Cr}}^{3+}\phantom{\rule{0.3em}{0ex}}(S=3\text{∕}2)$ substituted for the low spin ${\mathrm{Co}}_{\text{oct}}^{3+}\phantom{\rule{0.3em}{0ex}}(S=0)$ greatly affect the magnetic properties. Both coupling constants of the antiferromagnetic (interchain) and ferromagnetic (intrachain) interactions decrease as $x$ increases up to $x=0.10$ in ${\text{Ca}}_{3}{\text{Co}}_{2\ensuremath{-}x}{\text{Cr}}_{x}{\mathrm{O}}_{6}$. This is reflected by the decrease of the magnetization saturation and of the magnetic field characteristic of the ferri- to ferromagnetic transition, below the N\'eel temperature. These results are interpreted on the basis of an antiferromagnetic coupling between ${\mathrm{Co}}_{\text{oct}}^{3+}$ and its high spin ${\mathrm{Co}}_{\text{TP}}^{3+}$ neighbours. Furthermore, the magnetization jumps, observed below $Tl10\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, tend to be suppressed on the isothermal magnetization curves as soon as $x=0.05$. These results emphasize the great sensitivity of the ${\text{Ca}}_{3}{\text{Co}}_{2}{\mathrm{O}}_{6}$ magnetic behavior to the presence of a magnetic foreign cation, with, in particular, a ${T}_{N}$ decrease probed by specific-heat measurements.