Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent random variables, and $η$ be a counting random variable independent of this sequence. In addition, let $S_0:=0$ and $S_n:=ξ_1+ξ_2+\cdots+ξ_n$ for $n\geqslant1$. We consider conditions for random variables $\{ξ_1,ξ_2,\ldots\}$ and $η$ under which the distribution functions of the random maximum $ξ_{(η)}:=\max\{0,ξ_1,ξ_2,\ldots,ξ_η\}$ and of the random maximum of sums $S_{(η)}:=\max\{S_0,S_1,S_2,\ldots,S_η\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{ξ_1,ξ_2,\ldots\}$ are not necessarily identically distributed.