In 2012 Karapınar and Samet introduced the notion of α- contractive mappings and proved some well-known fixed point theorems. On the other hand in 2006 Mustafa and Sims gave the definition of G-metric space which is the generalization of metric space. In 2013 Algamdi and Karapınar combined these two notions as G-β - contractive mappings. They presented and proved related fixed point theorems. In this paper β-admissible mappings in G-metric spaces are restricted and some fixed point theorems are given. Our main results will give a new perspective to previous works in G-metrics. Indeed, as a result of a special selection of the given mappings, we obtained the main results of Algamdi and Karapınar as a conclusion