Polycyclic aromatic hydrocarbons (PAHs) are of great astrochemical and astrobiological interest due to their potential to form prebiotic molecules. We analyse the 7.7 and 8.6{mu}m PAH bands in 126 pre-dominantly starburst- dominated galaxies extracted from the Spitzer/IRS ATLAS project. Based on the peak positions of these bands, we classify them into the different A, B, and C Peeters' classes, which allows us to address the potential characteristics of the PAH-emitting population. We compare this analysis with previous work focused on the 6.2{mu}m PAH band for the same sample. For the first time in the literature, this statistical analysis is performed on a sample of galaxies. In our sample, the 7.7{mu}m complex is equally distributed in A and B object's class while the 8.6{mu}m band presents more class B sources. Moreover, 39 per cent of the galaxies were distributed into A class objects for both 6.2 and 7.7 {mu}m bands and only 18 per cent received the same A classification for the three bands. The 'A A A' galaxies presented higher temperatures and less dust in their interstellar medium. Considering the redshift range covered by our sample, the distribution of the three bands into the different Peeters' classes reveals a potential cosmological evolution in the molecular nature of the PAHs that dominate the interstellar medium in these galaxies, where B class objects seem to be more frequent at higher redshifts and, therefore, further studies have to be addressed.