Catalyzes the phosphorylation of LysW-gamma-alpha-aminoadipate. Does not phosphorylate N-acetyl-glutamate. [amino-group carrier protein]-C-terminal-N-(1,4-dicarboxybutan-1-yl)-L-glutamine + ATP = [amino-group carrier protein]-C-terminal-N-(1-carboxy-5-phosphooxy-5-oxopentan-1-yl)-L-glutamine + ADP Amino-acid biosynthesis; L-lysine biosynthesis via AAA pathway; L-lysine from L-alpha-aminoadipate (Thermus route): step 2/5. Belongs to the acetylglutamate kinase family. LysZ subfamily.