Fremantle
Is this the old Fremantle weather station? It's in the council depot on Knutsford Street. The BOM says that the Fremantle station (site number 9017, which ran from 1852 to 1992) was at -32.0550° 115.7500° (see box on the map below). That's down near the old courthouse in Henderson Street, so I guess it probably was moved around over the years.

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