HCl/H2O with alkene: Which one reacts?
I think the last step  is for the Hydrolysis.  After reaction with $\ce{LiAlH_4}$, the amide is reduced to amine and thus when it is reacted with a ketone it forms an imine($1G$) like this below,

which quickly tautomerises to form the most stable Enamine as this  

After  that when it is reacted with $\ce{CH_3Br}$, the nitrogen's lone pair conjugates and the nucleophilic attack occurs from the adjacent carbon(like Michael addition) with attached methyl group. Thus you have the intermidiate substituted proiduct as, 

But this imine  is hydrolysed in the presence of $\ce{H^+/H_2O}$ to give you back the original ketone but with an extra methyl group attached to it. So I think your final product $1H$ should be this, 

and also you get the amine after hydrolysis which is actually the  same as $1F$.