To add to the contracting companies (thank you Occam for providing me this well detailed information)
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Weir Marine - https://www.global.weir/businesses/weir-marine-engineering-(nete) - operates Naval Engineering Test Establishment (NETE), based in Montreal but has offices on both coasts within the dockyards. NETE provides a wide range of technical services to DND. For example, all the equipment we plan on putting on the ships has to be subjected to "shake and bake" testing (shock, vibration and environmental). They're embedded contractors, both at DGMEPM in Ottawa and at the Fleet Maintenance Facilities on the coasts.
Valcom Consulting - https://valcom.ca/ - they're mostly headhunters; they provide a lot of contractor support in the ADM(IM) and ADM(Mat) groups. When I was at 76 Comm Gp in Ottawa, there were dozens of 2nd and 3rd line tech support contractors on the DWAN and CSNI networks. Valcom also provides some specialized engineering people in ADM(Mat), where I'm at now - mostly support to LCMMs and Project Managers. We have a couple of former CF techs in our section that are contractors from Valcom. Valcom also has a support contract to issue our RFQs and collate responses from industry for all our procurements under $25K at DGMEPM - not sure who the Army and Air Force use. Valcom has a bunch of positions advertised with DND right now.
Fleetway - http://www.fleetway.ca/ - Fleetway may not be an embedded contractor, I'm unsure if they have any embedded positions on the coasts. Fleetway is the Navy's Technical Data Management provider. They produce, update and track the Technical Data Packages for every system on the HALIFAX Class ships. If I remove a system from the ships, and replace it with another, Fleetway updates the drawings, cable run sheets, data lists, reference publication data, etc. They are also tasked to draft Engineering Change specifications to install/change/remove systems from the ships. They have @fleetway.ca e-mail addresses, or at least the ones I know do. Fleetway has a bunch of jobs advertised right now that are targeted (mostly) to former sailors. Fleetway is a subsidiary of Irving Shipbuilding, who will be building the AOPS and CSC classes of ship.
SNC Lavalin - http://www.snclavalin.com/en/infrastructure - SNC Lavalin is the In Service Support Contractor for the KINGSTON class ships. Their engineers and technicians carry out most of the maintenance on that class, as well as maintain the TDP like Fleetway does above. I wouldn't call them embedded contractors, as I believe they have @snclavalin.ca e-mail addresses, but they do have offices within the dockyards east and west as, well, they're the ones doing the maintenance on the MCDVs. SNC Lavalin also has contracts to maintain aeronautical navigation aids at several DND airfields across Canada - Gander I think is one, Moose Jaw as well if I remember correctly. No ATIS Techs at those airfields.