We consider wireless sensor networks secured by the heterogeneous random key predistribution scheme under an on/off channel model. The heterogeneous random key predistribution scheme considers the case when the network includes sensor nodes with varying levels of resources, features, or connectivity requirements; e.g., regular nodes vs. cluster heads, but does not incorporate the fact that wireless channel are unreliable. To capture the unreliability of the wireless medium, we use an on/off channel model; wherein, each wireless channel is either on (with probability $α$) or off (with probability $1-α$) independently. We present conditions (in the form of zero-one laws) on how to scale the parameters of the network model so that with high probability the network is $k$-connected, i.e., the network remains connected even if any $k-1$ nodes fail or leave the network. We also present numerical results to support these conditions in the finite-node regime.