  The MnBi$_{2}$Te$_{4}$ family has attracted significant attention due to its
rich topological states such as the quantum anomalous Hall (QAH) insulator
state, the axion insulator state, and the magnetic Weyl semimetal state.
Nevertheless, the intrinsic antiferromagnetic (AFM) interlayer coupling in
MnBi$_{2}$Te$_{4}$ partly hinders the realization of "high-temperature" QAH
effect. Here, by using first-principles electronic structure calculations, we
design a new class of materials Mn\emph{X}$_{2}$\emph{B}$_{2}$\emph{T}$_{6}$
(\emph{X}=Ge, Sn, or Pb; \emph{B}=Sb or Bi; \emph{T}=Se or Te) based on the
\emph{X}$_{2}$\emph{B}$_{2}$\emph{T}$_{5}$ structures rather than the
Bi$_{2}$Te$_{3}$ family. We find that each septuple-layer
Mn\emph{B}$_{2}$\emph{T}$_{4}$ is sandwiched by two [\emph{X}\emph{T}] layers,
which may turn the AFM interlayer coupling into a ferromagnetic (FM) coupling.
The calculations specifically demonstrate that
\emph{MnGe}$_{2}$\emph{Sb}$_{2}$\emph{Te}$_{6}$,
\emph{MnGe}$_{2}$\emph{Bi}$_{2}$\emph{Te}$_{6}$, and
\emph{MnPb}$_{2}$\emph{Bi}$_{2}$\emph{Te}$_{6}$ are FM axion insulators, while
MnGe$_{2}$Sb$_{2}$Se$_{6}$, MnGe$_{2}$Bi$_{2}$Se$_{6}$,
MnSn$_{2}$Sb$_{2}$Te$_{6}$, and MnSn$_{2}$Bi$_{2}$Te$_{6}$ are A-type AFM axion
insulators. These seven materials all have an out-of-plane easy axis of
magnetization. The Mn\emph{X}$_{2}$\emph{B}$_{2}$\emph{T}$_{6}$ family thus
offers a promising platform beyond the MnBi$_{2}$Te$_{4}$ family for the
realization of quantized magnetoelectric effect and "high-temperature" QAH
effect in future experiments.
