core language - Functions that creates symbol in a specific context

I'm trying to make a function f that takes a symbol, a value, and a context, and creates the symbol in the requested context and assigns it the passed value. The obvious doesn't work. I.e.:

Attributes[f]={HoldAllComplete};
f[symbol_,value_,context_]:=(Begin[context];Set[symbol,value];End[])

Creates the symbol in the current context, as opposed to the one that is passed as an argument. This seems to happen because the FrontEnd, as soon as it sees f getting evaluated, creates the symbol in the current context despite HoldAllComplete. I tried adding Remove[symbol] to the function but that doesn't work. Any thoughts?

Answer

The problem is that if you pass a symbol, it will be created already during the parsing stage, when you pass it, in the current context. Therefore I suggest to pass its string name instead. This function will do the job:

ClearAll[f];
f[symbolName_String, value_, context_] :=
  Block[{$ContextPath},
    BeginPackage[context];
    ToExpression[
       symbolName, 
       StandardForm, 
       Function[name, Set[name, value], HoldAll]

    ];
    EndPackage[]
  ] 

for example

f["a", 10, "Test`"]
Test`a

(* 10 *)

If you still want to pass a symbol, you can use this:

ClearAll[f];
f[symbol_, value_, context_] :=
  With[{set = MakeBoxes[symbol = value]},
    Block[{$ContextPath},
      BeginPackage[context];
      ReleaseHold[MakeExpression@set];
      EndPackage[]]];

which is a version of the code I used here. But be aware that you will also create the symbol symbol in the current working context, so you may additionally use Remove to remove it.

So, for example:

f[a, 20, "Test`"]
Test`a

(* 20 *)