For the Internet of Things (IoT), there might be a large number of devices to be connected to the Internet through wireless technologies. In general, IoT devices would have various constraints due to limited processing capability, memory, energy source, and so on, and it is desirable to employ efficient wireless transmission schemes, especially for uplink transmissions. For example, orthogonal frequency division multiplexing (OFDM) with index modulation (IM) or OFDM-IM can be considered for IoT devices due to its energy efficiency. In this paper, we study a different IM scheme for a single-carrier (SC) system, which is referred to as SCIM. While SCIM is similar to OFDM-IM in terms of energy efficiency, SCIM may be better suited for IoT uplink because it has a low peak-to-average power ratio (PAPR) and does not require inverse fast Fourier transform (FFT) at devices compared to OFDM-IM. We also consider precoding for SCIM and generalize it to multiple access channel so that multiple IoT devices can share the same radio resource block. To detect precoded SCIM signals, low-complexity detectors are derived. For a better performance, based on variational inference that is widely used in machine learning, we consider a detector that provides an approximate solution to an optimal detection.