Let K be an abelian group of order v. A Steiner quadruple system of order v (SQS(v)) (K,B) is called symmetric K-invariant if for each B∈B, it holds that B+x∈B for each x∈K and B=−B+y for some y∈K. When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K-invariant 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 SQS(v) shown in Piotrowski's thesis in 1985. In this paper, we prove that a symmetric K-invariant SQS(v) exists if and only if v≡2,4(mod6), the order of each element of K is not divisible by 8, and there exists a symmetric cyclic SQS(2p) for any odd prime divisor p of v.