Using the Hartree approximation, the $8\ifmmode\times\else\texttimes\fi{}8$ Kane Hamiltonian, and the envelope-function scheme the electronic structure of electrons bound within an inversion layer on p-InAs in a Mosfet geometry is computed self-consistently and studied as a function of the two-dimensional electron density ${N}_{S}$ and the doping concentration ${N}_{A}{\ensuremath{-}N}_{D}.$ The subband spin splitting ${\ensuremath{\delta}}_{k}^{\ensuremath{\nu}}$ at an in-plane wave vector k varies almost linearly with ${N}_{S},$ and for the same electron density and k it is larger in the lower subbands. Likewise, the k-dependent subband Rashba parameter ${\ensuremath{\alpha}}_{k}^{\ensuremath{\nu}}$ at a given k shows an analogous behavior. Varying the doping concentration in the interval $1.8\ifmmode\times\else\texttimes\fi{}{10}^{15}--1.8\ifmmode\times\else\texttimes\fi{}{10}^{17}{\mathrm{cm}}^{\ensuremath{-}3},$ in subband \ensuremath{\nu} the spin-splitting ${\ensuremath{\delta}}_{\ensuremath{\nu}}$ at the Fermi level is computed for ${N}_{S}$ in the range ${10}^{11}--4.8\ifmmode\times\else\texttimes\fi{}{10}^{12}{\mathrm{cm}}^{\ensuremath{-}2},$ where it is found to be an increasing function of ${N}_{S};$ moreover, it is largest in the ground subband. At the Fermi level, the corresponding Rashba parameter ${\ensuremath{\alpha}}_{\ensuremath{\nu}}$ is also computed as a function of ${N}_{S}$ in both the ground and first excited subbands while ${N}_{A}{\ensuremath{-}N}_{D}$ is varied from $0.433\ifmmode\times\else\texttimes\fi{}{10}^{17}$ to $1.8\ifmmode\times\else\texttimes\fi{}{10}^{17}{\mathrm{cm}}^{\ensuremath{-}3}.$ In this range, whereas ${\ensuremath{\alpha}}_{1}$ simply shows a decreasing trend as a function of ${N}_{S},$ ${\ensuremath{\alpha}}_{0}$ exhibits new and counterintuitive ${N}_{S}$ dependencies as ${N}_{A}{\ensuremath{-}N}_{D}$ is varied. Moreover, ${\ensuremath{\alpha}}_{1}$ can either be larger or smaller than ${\ensuremath{\alpha}}_{0},$ even when tunneling into the barrier is completely neglected. In addition, the role of the first excited subband on the overall ${N}_{S}$ dependence of ${\ensuremath{\alpha}}_{0}$ turns out to be crucial.