Let K be an abelian group of order v. A Steiner quadruple system of order v ($\text{SQS}(v)$) $(K,\mathcal{B})$ is called symmetric K-invariant if for each $B\in \mathcal{B}$, it holds that $B+x\in \mathcal{B}$ for each $x\in K$ and $B=-B+y$ for some $y\in K$. When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K-invariant $\text{SQS}(v)$ was given by Munemasa and Sawa, which is a generalization of a necessary and sufficient condition for the existence of a symmetric cyclic $\text{SQS}(v)$ shown in Piotrowski's thesis in 1985. In this paper, we prove that a symmetric K-invariant $\text{SQS}(v)$ exists if and only if $v\equiv 2,4(\mathrm{mod}6)$, the order of each element of K is not divisible by 8, and there exists a symmetric cyclic $\text{SQS}(2p)$ for any odd prime divisor p of v.

令K为v的阿贝尔群。阶v的Steiner四元系统（$\text{SQS}（v）$） $（ķ，\mathcal{乙}）$称为对称K-不变$乙\in \mathcal{乙}$，它认为 $乙+X\in \mathcal{乙}$ 每个 $X\in ķ$ 和 $乙=-乙+ÿ$ 对于一些 $ÿ\in ķ$。当K的Sylow 2子群是循环的时，存在对称K不变量的充要条件$\text{SQS}（v）$ 由Munemasa和Sawa给出，它是对称环存在的必要条件的充分概括 $\text{SQS}（v）$在1985年的Piotrowski论文中得到证明。在本文中，我们证明了对称K-不变量$\text{SQS}（v）$ 存在且仅当 $v\equiv 2，4（\mathrm{模}6）$，K的每个元素的顺序不能被8整除，并且存在一个对称的循环$\text{SQS}（2p）$对于任何奇素因子p的v。