A multiphoton ionization process of He in intense extreme ultraviolet free-electron laser fields is investigated in a photon energy range of $h\mathit{v}=20.1\ensuremath{-}20.8\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ using shot-by-shot photoelectron spectroscopy. A photoelectron peak resulting from sequential double ionization of He due to five-photon absorption is observed around $h\mathit{v}=20.4\phantom{\rule{0.16em}{0ex}}$. Shot-by-shot analysis of photoelectron spectra shows that two-photon resonance to the $\mathrm{H}{\mathrm{e}}^{+}(2s{\phantom{\rule{0.16em}{0ex}}}^{2}S)$ state from the ${\mathrm{He}}^{+}$ ground state is responsible for the substantial enhancement of five-photon double ionization. Effective cross sections for two-photon absorption between the $1s$ and $2s$ states of ${\mathrm{He}}^{+}$ and for one-photon absorption from the ${\mathrm{He}}^{+}$ $2s$ state are obtained.