  Recently, the evidence for gamma-ray emission has been found in the
$Fermi$-LAT observation for the outer halo of Andromeda galaxy (M31). The dark
matter (DM) annihilation offers a possible explanation on the gamma-ray
radiation. In this work, we focus on the dark matter annihilation within
minispikes around intermediate-mass black holes (IMBHs) with masses ranging
from $100~\mathrm{M_\odot}$ to $10^6~\mathrm{M_\odot}$. When the thermal
annihilation relic cross section $\left\langle \sigma v \right\rangle = 3
\times 10^{-26}~\mathrm {cm} ^{3}\;\mathrm {s} ^{-1}$ is adopted, we conduct an
investigation on the population of IMBHs in the spherical halo area of M31. We
find that there could be more than 65 IMBHs with masses of $ 100~
\mathrm{M_\odot}$ surrounded by the DM minispikes as the remnants of Population
III stars in the M31 spherical halo, and it is almost impossible for the
existence of minspikes around IMBHs with masses above $10^4~ \mathrm{M_\odot}$
which could be formed by the collapse of primordial cold gas, for both dark
matter annihilation channels $b\bar{b}$ and $\tau^{+}\tau^{-}$. The properties
of dark matter have been further explored with the simulation of these two
scenarios for IMBHs formation.
