The account used with a template is the account set when the template is created and stored in that template. If your group mail account has a signature associated with it, that signature will be added, however if you want to retain the default signature on your group account and create a message from your template without it, then you need to create the message and remove the signature e.g.
Sub Quickturnaround()
Dim oItem As MailItem
Dim oDoc As Object
Dim oBm As Object
Dim oRng As Object
'Open Quick Turnaround email template
Set oItem = Application.CreateItemFromTemplate( _
"C:\Users\kyle-p\AppData\Roaming\" _
& "Microsoft\Templates\Quick Turnaround.oft")
Set oDoc = oItem.GetInspector.WordEditor
oItem.Display
Set oBm = oDoc.Bookmarks("_MailAutoSig")
If Not oBm Is Nothing Then
Set oRng = oBm.Range
oRng.Text = ""
End If
lbl_Exit:
Set oItem = Nothing
Set oDoc = Nothing
Set oBm = Nothing
Set oRng = Nothing
Exit Sub
End Sub