Using a large collection of near-infrared spectra (2.5-5.4 μm) of Galactic H II regions and H II region-like objects, we perform a systematic investigation of astronomical polycyclic aromatic hydrocarbon (PAH) features. Thirty-six objects were observed using the infrared camera on board the AKARI satellite as a part of a director's time program. In addition to the well known 3.3-3.6 μm features, most spectra show a relatively weak emission feature at 5.22 μm with sufficient signal-to-noise ratios, which we identify as the PAH 5.25 μm band (previously reported). By careful analysis, we find good correlations between the 5.25 μm band and both the aromatic hydrocarbon feature at 3.3 μm and the aliphatic hydrocarbon features at around 3.4-3.6 μm. The present results give us convincing evidence that the astronomical 5.25 μm band is associated with C-H vibrations, as suggested by previous studies, and show its potential to probe the PAH size distribution. The analysis also shows that the aliphatic-to-aromatic ratio of I 3.4-3.6 μm/I 3.3 μm decreases against the ratio of the 3.7 μm continuum intensity to the 3.3 μm band, I cont, 3.7 μm/I 3.3 μm, which is an indicator of the ionization fraction of PAHs. The midinfrared color of I 9 μm/I 18 μm also declines steeply against the ratio of the hydrogen recombination line Brα at 4.05 μm to the 3.3 μm band, I Brα/I 3.3 μm. These facts indicate possible dust processing inside or at the boundary of ionized gas.