This paper finds that agentic AI systems intensify existing challenges to traditional public sector oversight mechanisms — which rely on siloed compliance units and episodic approvals rather than continuous, integrated supervision. We identify five governance dimensions essential for responsible agent deployment: cross-departmental implementation, comprehensive evaluation, enhanced security protocols, operational visibility, and systematic auditing. We evaluate the capacity of existing oversight structures to meet these challenges, via a mixed-methods approach consisting of a literature review and interviews with civil servants in AI-related roles. We find that agent oversight poses intensified versions of three existing governance challenges: continuous oversight, deeper integration of governance and operational capabilities, and interdepartmental coordination. We propose approaches that both adapt institutional mechanisms and design agent architectures compatible with public sector constraints.