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Designing safe and predictable software for embedded applications is a considerable challenge, which has been addressed by the COMDES-II framework. This is a component-based software framework employing a timed multitasking model of computation supported by a run-time environment. This paper presents a real-time kernel implementing the run-time environment - the HARTEXmu kernel that has been specifically developed for COMDES-II systems. The design and implementation of the kernel take into account the limitations of embedded systems, e.g. a small memory footprint and reduced kernel overhead. It also features Boolean vector processing that eliminates the need for linked-list queues, resulting in constant execution time of kernel functions independent of the number of tasks involved.