For a group action ${alpha}$ on a Kac algebra $mathbb{K}$ with the crossed product Kac algebra $mathbb{K}{ times}_{alpha}G$, we will show that ${pi}_{alpha}(mathbb{K})$ is a sub-Kac algebra of $mathbb{K}{ times}_{alpha}G$. We will also investigate the intrinsic group $G(mathbb{K})$ of $mathbb{K}$ and get a group action ${eta}$ on a symmetric Kac algebra $mathbb{K}_s(G(mathbb{K})$ with the crossed product sub-Kac algebra $mathbb{K}_s(G(mathbb{K}){ times}_{eta}G$ of $mathbb{K}{ times}_{alpha}G$.