This is a non-exhaustive list of factors influencing the permafrost thickness :
- material type
- nature of the regional climate (dry, oceans, continental)
- geologic history of the permafrost (formed after the last glacial maximum, or before?)
Time is important: older permafrost can be very deep, especially in areas such as in Siberia, Yukon/Alaska where the effect of glacier was limited, thus exposing the ground to cold conditions for long time. Some areas in the Canadian archipelago are in excess of 1 km deep, and more than 500 m is not uncommon, largely depending on local and time conditions.
Yet, huge discrepancies in permafrost depth can be found between western Canada/AK, eastern Canada, Northern Europe and Siberia, given a constant latitude, and therefore isolation and thus comparable aspect. As an example at Svalbard (78N), permafrost depth was reported to be less than 400 m, while regional depth in the central canadian archipelago north of Baffin Island is more than 500 m (for 73 - 75 N) and 1500 m in northern Siberia (which is at a lower latitude than Svalbard).
To get back to your question, aspect (or solar radiation input) is pretty much uniform for a given latitude - but in this case, for the same latitude, permafrost depth vary.
On the other hand, material type vary locally. If wetlands, mosses and vegetation will contribute to cool the ground, enabling deeper permafrost in those areas. Areas were there is a lot of snow (snowbanks) will have a warmer permafrost compared to crests nearby. Lakes change how heat can be extracted from the ground (latent heat effect) so permafrost will be thin under a lake compared to around the lake (lake being a type of material in the context of your question). Desert ? Rock outcrop ?
Each type of environment behave in a certain way regarding heat extraction and ultimately heat exchanges, which have a lot to do with the nature of the surface..