Was one of the fist things I mapped out. Some of them are less intuitive than others and required some discussion (Artisan, Forgery, Armorer, Animal Handling, Heavy Weapons) and I feel there are still some wiggle room there (Heavy Weapons such as machine guns and auto cannons might be Firearms rather than Exotic and Artisan might be Engineering rather than Con, for example and you can possible also make an argument that forgery when used to create electronic devices such as fake credsticks would use Electronics rather than Con...) while others required some digging before I found a reference (such as Chemistry) and then you have skills that have been deliberately taken out (Binding... also Arcane?).
SR5 Agility linked Skills:
Archery = Athletics
Automatics = Firearms
Blades = Close Combat
Clubs = Close Combat
Escape Artist = Athletics
Exotic Melee = Exotic Weapons (whips)
Exotic Ranged = Exotic Weapons (grenade/missile launchers, lasers, flamethrowers, squirt, dart...)
Gunnery = Engineering
Gymnastics = Athletics
Heavy Weapons = Exotic Weapons? Firearms?
Locksmith = Engineering
Longarms = Firearms
Palming = Stealth
Pistols = Firearms
Sneaking = Stealth
Throwing Weapons = Athletics
Unarmed Combat = Close Combat
SR5 Body linked Skills:
Diving = Athletics
Free Fall = Athletics
SR5 Reaction linked Skills:
Pilot Aerospace = Piloting
Pilot Aircraft = Piloting
Pilot Walker = Piloting
Pilot Exotic Vehicle = Piloting
Pilot Ground Craft = Piloting
Pilot Watercraft = Piloting
SR5 Strength linked Skills:
Running = Athletics
Swimming = Athletics
SR5 Charisma linked Skills:
Animal Handling = Influence
Con = Con
Etiquette = Influence
Impersonation = Con
Instruction = Influence
Intimidation = Influence
Leadership = Influence
Negotiation = Influence
Performance = Con
SR5 Intuition linked Skills:
Artisan = Con?
Assensing = Astral
Disguise = Con
Navigation = Outdoors
Perception = Perception
Tracking = Outdoors
SR5 Logic linked Skills:
Aeronautics Mechanics = Engineering
Arcane = Astral?
Armorer = Engineering
Automotive Mechanic = Engineering
Biotechnology = Biotech
Chemistry = Biotech
Computer = Electronics
Cybertechnology = Biotech
Cybercombat = Cracking
Demolitions = Engineering
Electronic Warfare = Cracking
First Aid = Biotech
Industrial Mechanics = Engineering
Hacking = Cracking
Hardware = Electronics
Medicine = Biotech
Nautical Mechanics = Engineering
Software = Electronics
Forgery = Con?
SR5 Willpower linked Skills:
Astral Combat = Astral
Survival = Outdoors
SR5 Magic linked Skills:
Alchemy = Enchanting
Artificing = Enchanting
Banishing = Conjuring
Binding = ?
Counterspelling = Sorcery
Disenchanting = Enchanting
Ritual Spellcasting = Sorcery
Spellcasting = Sorcery
Summoning = Conjuring
SR5 Resonance linked Skills:
Compiling = Tasking
Decompiling = Tasking
Registering = Tasking
Grouped by SR6 skills:
Athletics
Archery, Escape Artist, Gymnastics, Throwing Weapons, Diving, Free Fall, Running, Swimming
Biotech*
Biotechnology, Chemistry, Cybertechnology, First Aid, Medicine
Close Combat
Blades, Clubs, Unarmed Combat
Con (Acting, Disguise, Impersonation, Performance)
Con, Impersonation, Performance, Artisan, Disguise, Forgery
Cracking*
Cybercombat, Electronic Warfare, Hacking
Electronic (no longer used for repairing matrix damage!)
Computer, Hardware, Software
Engineering (also used for locksmith, demolitions, repairing matrix damage and gunnery)
Gunnery, Locksmith, Aeronautics Mechanic, Armorer, Automotive Mechanic, Demolitions, Industrial Mechanic, Nautical Mechanic
Exotic Weapons*
Must pick specialization, once per weapon, and can pick specialization several times.
Firearms (Automatics, Longarms, Pistols, Rifles, Shotguns)
Automatics, Longarms, Pistols
Influence
Animal Handling, Etiquette, Instruction, Intimidation, Leadership, Negotiation
Outdoors (Navigation, Survival, Tracking, by environment (Woods, Desert, Urban Areas, etc.))
Navigation, Survival, Tracking
Perception (Visual, Aural, Tactile, by Environment (Woods, Desert, Urban, etc.))
Perception
Piloting (specializations are now just Ground Craft, Aircraft, Watercraft)
Aerospace, Aircraft, Walker, Exotic, Ground craft, Watercraft
Stealth (also mentions disguise which will be renamed to camouflage)
Palming, Sneaking
Astral* (Astral Combat, Astral Signatures, Emotional States, Spirit Types)
Assensing, Arcane?, Astral Combat
Conjuring* (no longer include Binding!)
Banishing, Summoning
Enchanting*
Alchemy, Artificing, Disenchanting
Tasking*
Compiling, Decompiling, Registering