HHO (hybrid high-order) is a hybrid finite element scheme in the sense that it approximates a solution using polynomial degrees of freedom on both cells and faces of a mesh. Some of the main advantages are that general polyhedral meshes and high order approximation are supported without additional complexity. Multigrid methods are solvers for the linear systems arising from the discretization of PDE (partial diferential equation). Their development exploits information about the equation and the discretization, and they can therefore outperform general-use solvers. Heuristically, a multigrid method solves the PDE on a sequence of levels simultaneously, ranging from the finest level, which defines the problem we actually want to solve, to coarser "help" levels, where approximate corrections to the fine-level solution can be obtained more easily. The most common multigrid method is the $h$-multigrid where coarser levels are obtained using coarser meshes. An alternative is the $p$-multigrid where coarser levels are obtained by lowering the polynomial degree in a finite element scheme. The advantage of the $p$-multigrid is that it avoids the cumbersome construction of a sequence of linked meshes which is inherent to the $h$-multigrid. This is especially so in the case of hybrid methods, where degrees of freedom live on faces and mesh coarsening or refinement must be carried out with respect to the skeleton \cite{Paper1}. $P$-multigrids have previously been studied numerically for hybrid methods, with positive results \cite{Paper2}. Theoretically, however, the only existing works have addressed DG \cite{Paper3} and VEM \cite{Paper4}. In the present paper we study a nested inherited $p$-multigrid algorithm for the Poisson problem in HHO formulation. We show that the algorithm converges uniformly with respect to mesh size and with only a small polynomial deterioration with respect to the degree.