  Magnetic topological insulators provide a platform for emergent phenomena
arising from the interplay between magnetism and band topology. Here we report
the single crystal growth, crystal structure, magnetic and transport
properties, as well as the neutron scattering studies of topological insulator
series (Ge$_{1-\delta-x}$Mn$_x$)$_2$Bi$_2$Te$_5$ ($x\leq 0.47$, $0.11 \leq
\delta \leq 0.20$). Upon doping up to $x = 0.47$, the lattice parameter $c$
decreases by 0.8\%, while the lattice parameter $a$ remains nearly unchanged.
Significant Ge vacancies and Ge/Bi site mixing are revealed via elemental
analysis as well as refinements of the neutron and X-ray diffraction data,
resulting in holes dominating the charge transport. At $x = 0.47$, below 10.8
K, a bilayer A-type antiferromagnetic ordered state emerges, featuring an
ordered moment of 3.0(3) $\mu_{B}$/Mn at 5 K, with the $c$ axis as the easy
axis. Magnetization data unveil a much stronger interlayer antiferromagnetic
exchange interaction and a much smaller uniaxial anisotropy compared to
MnBi$_{2}$Te$_{4}$. We attribute the former to the shorter superexchange path
and the latter to the smaller ligand-field splitting in
(Ge$_{1-\delta-x}$Mn$_x$)$_2$Bi$_2$Te$_5$. Our study demonstrates that this
series of materials holds promise for the investigation of the Layer Hall
effect and quantum metric nonlinear Hall effect.
