This article presents a structured framework for Human-AI collaboration in Security Operations Centers (SOCs), integrating AI autonomy, trust calibration, and Human-in-the-loop decision making. Existing frameworks in SOCs often focus narrowly on automation, lacking systematic structures to manage human oversight, trust calibration, and scalable autonomy with AI. Many assume static or binary autonomy settings, failing to account for the varied complexity, criticality, and risk across SOC tasks considering Humans and AI collaboration. To address these limitations, we propose a novel autonomy tiered framework grounded in five levels of AI autonomy from manual to fully autonomous, mapped to Human-in-the-Loop (HITL) roles and task-specific trust thresholds. This enables adaptive and explainable AI integration across core SOC functions, including monitoring, protection, threat detection, alert triage, and incident response. The proposed framework differentiates itself from previous research by creating formal connections between autonomy, trust, and HITL across various SOC levels, which allows for adaptive task distribution according to operational complexity and associated risks. The framework is exemplified through a simulated cyber range that features the cybersecurity AI-Avatar, a fine-tuned LLM-based SOC assistant. The AI-Avatar case study illustrates human-AI collaboration for SOC tasks, reducing alert fatigue, enhancing response coordination, and strategically calibrating trust. This research systematically presents both the theoretical and practical aspects and feasibility of designing next-generation cognitive SOCs that leverage AI not to replace but to enhance human decision-making.