Outer function for nested list

Question

I have the following routine where I build a pair of function name lists {Flist, Klist} and 1D argument list {a1,a2,a3,a4,a5}:

T = Transpose@{ToExpression@ Table[StringJoin["a", ToString[i]], {i, 1, 5}]};
Flist = {F1, F2, F3};
Klist = {K1, K2, K3};
Flatten[Outer[{#1, #2} &, {Flist, Klist}, T], 3]

Out[149]= {{F1, a1}, {F1, a2}, {F1, a3}, {F1, a4}, {F1, a5},
{F2, a1}, {F2, a2}, {F2, a3}, {F2, a4}, {F2, a5},
{F3, a1}, {F3, a2}, {F3, a3}, {F3, a4}, {F3, a5},
{K1, a1}, {K1, a2}, {K1, a3}, {K1, a4}, {K1, a5},
{K2, a1}, {K2, a2}, {K2, a3}, {K2, a4}, {K2, a5},
{K3, a1}, {K3, a2}, {K3, a3}, {K3, a4}, {K3, a5}}

With the one argument or many, e.g. {Fn, arg1_n, arg2_n}, it works, but I am confused with the Outer function applied to the nested lists as to:

T = Transpose@{ToExpression@Table[StringJoin["a", ToString[i]], {i, 1, 5}],ToExpression@Table[StringJoin["b", ToString[i]], {i, 1, 5}]}

Out[151]= {{a1, b1}, {a2, b2}, {a3, b3}, {a4, b4}, {a5, b5}}

Flatten[Outer[{#1, #2} &, {Flist, Klist}, T, 1], 1]

Out[182]= {{{F1, F2, F3, F4, F5}, {a1, b1}}, {{F1, F2, F3, F4, 
   F5}, {a2, b2}}, {{F1, F2, F3, F4, F5}, {a3, b3}}, {{F1, F2, F3, F4,
    F5}, {a4, b4}}, {{F1, F2, F3, F4, F5}, {a5, b5}}, {{K1, K2, K3, 
   K4, K5}, {a1, b1}}, {{K1, K2, K3, K4, K5}, {a2, b2}}, {{K1, K2, K3,
    K4, K5}, {a3, b3}}, {{K1, K2, K3, K4, K5}, {a4, b4}}, {{K1, K2, 
   K3, K4, K5}, {a5, b5}}}

I played with dimensions allowed in Outer and Flatten and did not succeeded to obtain result in the following form:

{{F1, {a1, b1}}, {F1, {a2,b2}}, {F1, {a3,b3}},.......
{K3, {a3, b3}}, {K3, {a4, a4}}, {K3, {a5, b5}}}

Can someone advise how to obtained desired output having {Flist, Klist} and nested list T?

UPD Fr and Kr which @lericr asked below, was a typo in my code, which I fixed and recall them Klist and Klist. Sorry for omission.

Answer 1

Tuples may be easier than Outer. For example, using:

T = Transpose@{ToExpression@
     Table[StringJoin["a", ToString[i]], {i, 1, 5}], 
    ToExpression@Table[StringJoin["b", ToString[i]], {i, 1, 5}]};
Flist = {F1, F2, F3};
Klist = {K1, K2, K3};

you can do:

Tuples[{Flist~Join~Klist, T}]

which yields:

{{F1, {a1, b1}}, {F1, {a2, b2}}, {F1, {a3, b3}}, {F1, {a4, 
   b4}}, {F1, {a5, b5}}, {F2, {a1, b1}}, {F2, {a2, b2}}, {F2, {a3, 
   b3}}, {F2, {a4, b4}}, {F2, {a5, b5}}, {F3, {a1, b1}}, {F3, {a2, 
   b2}}, {F3, {a3, b3}}, {F3, {a4, b4}}, {F3, {a5, b5}}, {K1, {a1, 
   b1}}, {K1, {a2, b2}}, {K1, {a3, b3}}, {K1, {a4, b4}}, {K1, {a5, 
   b5}}, {K2, {a1, b1}}, {K2, {a2, b2}}, {K2, {a3, b3}}, {K2, {a4, 
   b4}}, {K2, {a5, b5}}, {K3, {a1, b1}}, {K3, {a2, b2}}, {K3, {a3, 
   b3}}, {K3, {a4, b4}}, {K3, {a5, b5}}}