In 1978, Byleen, Meakin and Pastijn[2,3] introduced the four - spiral semigroup Sp4 and the double four-spiral semigroup DSp4 and studied their properties in detail. These regular semigroups play an important role in the theory of idempotent generated bismple but not completely simple semigroups. The semigroup A(1, 2) was introduced as a tool to analyse the structure of DSp4 [3]. In 2003, Chandrasekaran and Loganathan[7] observed that A(1, 2) has an inverse transversal which is a bicyclic monoid and applied this fact to represent DSp4 as a regular Rees matrix semigroup over A(1, 2). This result is analogous to the representation due to Byleen [1] of the fundamental four spiral semigroup as a regular Rees matrix semigroup over the bicyclic monoid. Meakin [17] described the H-coextensions of Sp4, the fundamental four spiral semigroup analogous to Reilly)s description of the bisimple-w-semigroups[21]. So it is natural to ask the following question. Determine the H-coextensions of A(1, 2) and DSp4 (Problem 21, [7]). In this paper, first we describe the H-coextension of the R-unipotent semigroup A(1,2) and then we describe the H-coextension of the double four spiral semigroup DSp4.