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An increasing share of image and video content is analyzed by machines rather than viewed by humans, and therefore it becomes relevant to optimize codecs for such applications where the analysis is performed remotely. Unfortunately, conventional coding tools are challenging to specialize for machine tasks as they were originally designed for human perception. However, neural network based codecs can be jointly trained end-to-end with any convolutional neural network (CNN)-based task model. In this paper, we propose to study an end-to-end framework enabling efficient image compression for remote machine task analysis, using a chain composed of a compression module and a task algorithm that can be optimized end-to-end. We show that it is possible to significantly improve the task accuracy when fine-tuning jointly the codec and the task networks, especially at low bit-rates. Depending on training or deployment constraints, selective fine-tuning can be applied only on the encoder, decoder or task network and still achieve rate-accuracy improvements over an off-the-shelf codec and task network. Our results also demonstrate the flexibility of end-to-end pipelines for practical applications.