Issues with InverseFourierSequenceTransform?

Questions
1

Yes:

In[331]:= InverseFourierSequenceTransform[1, x, n, 
 FourierParameters -> {a, 1}]

Out[331]= (2 \[Pi])^((1 - a)/2) DiscreteDelta[n]

No:

In[329]:= InverseFourierSequenceTransform[1, x, n, 
 FourierParameters -> {a, 2 Pi}]

Out[329]= 0
2

And no:

In[1]:= InverseFourierSequenceTransform[1, x, n, FourierParameters -> {a, 1}]
Out[1]= InverseFourierSequenceTransform[1, x, {-0.00913271, 0.0914715}, 
                         FourierParameters -> {{1.95061, 1.19098}, 1}]

and so not yes

In[2]:= InverseFourierSequenceTransform[1, x, n, FourierParameters -> {a, 2 Pi}]
Out[2]= InverseFourierSequenceTransform[1, x, {-0.00913271, 0.0914715}, 
                             FourierParameters -> {{1.95061, 1.19098}, 2 \[Pi]}]

In[3]:= $Version
Out[3]= "10.4.1 for Microsoft Windows (64-bit) (April 11, 2016)"
3

Do this again with

In[18]:= $Version
Out[18]= "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"

telling in ref/InverseFourierSequenceTransform

InverseFourierSequenceTransform is the same as FourierCoefficient:

In[20]:= InverseFourierSequenceTransform[1, x, n, FourierParameters -> {1, 2 Pi}]
Out[20]= 0

In[21]:= FourierCoefficient[1, x, n, FourierParameters -> {1, 2 Pi}]
Out[21]= DiscreteDelta[n]

not really, but at least FourierCoefficient does not crack down under the FourierParameters given.