Properties of the Grain-128AEAD key re-introduction, as part of the cipher initialization, are analyzed and discussed. We consider and analyze several possible alternatives for key re-introduction and identify weaknesses, or potential weaknesses, in them. Our results show that it seems favorable to separate the state initialization, the key re-introduction, and the $A/R$ register initialization into three separate phases. Based on this, we propose a new cipher initialization and update the cipher version to Grain-128AEADv2. It can be noted that previously reported and published analysis of the cipher remains valid also for this new version.