A wireless device (S) includes a radio (R) for communicating via a wireless communication protocol which employs messages constructed as layer 2 MAC frames each including a layer 2 MAC header and a payload. The wireless device is configured to operate in (i) a first mode in which the wireless device transmits messages (N0, . . . , Nn) via the radio each including an IPv6 packet header and an upper layer protocol data unit encapsulated within the payload of a layer 2 MAC frame, and (ii) a second mode in which the wireless device transmits messages (M0, . . . , Mn) via the radio each including an upper layer protocol data unit encapsulated within the payload of a layer 2 MAC frame without including the IPv6 header. A relay device (T) comprises a radio for communicating via the wireless communication protocol which employs messages constructed as layer 2 MAC frames each including a layer 2 MAC header and a payload. In one power-saving aspect, the relay device is configured to perform a header insertion service (I) in which the relay device receives messages (M0, . . . , Mn) via its radio from the wireless device (S), each including an upper layer protocol data unit encapsulated within the payload of a layer 2 MAC frame without including the IPv6 header. The header insertion service inserts header information (A′) into the messages (M0, Mn) received from the wireless device (S) and retransmits the messages with complete headers as messages (M0′, . . . , Mn′). In another power-saving aspect, the relay device additionally or alternatively applies an acknowledgement detection criterion to filter out acknowledgements received from the server, and forwards the filtered out acknowledgements to the wireless device (S) via the radio at a reduced rate.