Final answer:
The Stalwarts, a faction within the Republican Party led by Roscoe Conkling, were the group that continued to support the patronage spoils system. They resisted civil service reform and were in direct opposition to the Half-Breeds, who supported some reform, and the Mugwumps, who broke away due to corruption issues.
Step-by-step explanation:
The group of Republicans who continued to support the patronage "spoils" system were the Stalwarts. This faction was led by Roscoe Conkling and strongly advocated the continuation of the system where political supporters are rewarded with government jobs. On the other hand, the Half-Breeds, led by James G. Blaine, supported some civil service reform, while the Mugwumps broke away from both the Stalwarts and Half-Breeds due to corruption. The Muckrakers, who are not part of this particular discourse, were reform-minded journalists in the early 20th century who aimed to expose social ills and corporate and political corruption.
The Stalwarts were opposed by the Half-Breeds within the Republican Party, the latter of whom called for a degree of civil service reform. By contrast, the Stalwarts staunchly defended the spoils system, resisting any such reforms. The political landscape of this time was a battle between maintaining traditional patronage practices and pushing for reform to create a more merit-based civil service.